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Gilbert Strang: Linear Algebra, Teaching, and MIT OpenCourseWare | Lex Fridman Podcast #52

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{'title': 'Gilbert Strang: Linear Algebra, Teaching, and MIT OpenCourseWare | Lex Fridman Podcast #52', 'heatmap': [{'end': 245.358, 'start': 203.951, 'weight': 1}], 'summary': "This podcast features mit professor gilbert strang's influential teachings in mathematics and their impact on ai, cash app's features and first charity sponsorship, matrix decomposition's visual appeal and math's role in youtube culture, the link between math and politics, expressive power of piecewise flat functions in engineering, and the significance of linear algebra in understanding high-dimensional surfaces and handling data in matrix form.", 'chapters': [{'end': 141.083, 'segs': [{'end': 28.323, 'src': 'embed', 'start': 0.069, 'weight': 0, 'content': [{'end': 2.751, 'text': 'The following is a conversation with Gilbert Strang.', 'start': 0.069, 'duration': 2.682}, {'end': 9.955, 'text': "He's a professor of mathematics at MIT and perhaps one of the most famous and impactful teachers of math in the world.", 'start': 3.411, 'duration': 6.544}, {'end': 15.259, 'text': 'His MIT OpenCourseWare lectures on linear algebra have been viewed millions of times.', 'start': 10.676, 'duration': 4.583}, {'end': 19.4, 'text': 'As an undergraduate student, I was one of those millions of students.', 'start': 16.018, 'duration': 3.382}, {'end': 22.341, 'text': "There's something inspiring about the way he teaches.", 'start': 19.98, 'duration': 2.361}, {'end': 28.323, 'text': "There's at once calm, simple, and yet full of passion for the elegance inherent to mathematics.", 'start': 22.781, 'duration': 5.542}], 'summary': 'Gilbert strang, a renowned mit math professor, has millions of views for his impactful linear algebra lectures.', 'duration': 28.254, 'max_score': 0.069, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk069.jpg'}, {'end': 72.473, 'src': 'embed', 'start': 47.973, 'weight': 1, 'content': [{'end': 56.724, 'text': 'From signals to images, from numerical optimization to robotics, computer vision, deep learning, computer graphics and everywhere outside AI,', 'start': 47.973, 'duration': 8.751}, {'end': 60.929, 'text': 'including, of course, a quantum mechanical study of our universe.', 'start': 56.724, 'duration': 4.205}, {'end': 64.131, 'text': 'This is the Artificial Intelligence Podcast.', 'start': 61.61, 'duration': 2.521}, {'end': 72.473, 'text': 'If you enjoy it, subscribe on YouTube, give it five stars on Apple Podcasts, support on Patreon or simply connect with me on Twitter.', 'start': 64.551, 'duration': 7.922}], 'summary': 'Artificial intelligence podcast covers a wide range of topics including computer vision, deep learning, and quantum mechanics.', 'duration': 24.5, 'max_score': 47.973, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk047973.jpg'}, {'end': 127.979, 'src': 'embed', 'start': 98.463, 'weight': 2, 'content': [{'end': 100.925, 'text': "ZipRecruiter is a tool that's already available for you.", 'start': 98.463, 'duration': 2.462}, {'end': 104.448, 'text': 'It seeks to make hiring simple, fast, and smart.', 'start': 101.285, 'duration': 3.163}, {'end': 112.154, 'text': 'For example, Codable co-founder Gretchen Huebner used ZipRecruiter to find a new game artist to join her education tech company.', 'start': 105.008, 'duration': 7.146}, {'end': 116.335, 'text': 'By using ZipRecruiter screening questions to filter candidates,', 'start': 112.714, 'duration': 3.621}, {'end': 125.178, 'text': 'Gretchen found it easier to focus on the best candidates and finally hiring the perfect person for the role in less than two weeks from start to finish.', 'start': 116.335, 'duration': 8.843}, {'end': 127.979, 'text': 'ZipRecruiter, the smartest way to hire.', 'start': 125.518, 'duration': 2.461}], 'summary': 'Ziprecruiter simplifies hiring, helping find and hire a game artist in less than two weeks using screening questions.', 'duration': 29.516, 'max_score': 98.463, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk098463.jpg'}], 'start': 0.069, 'title': 'Impact of mit professor on mathematics', 'summary': 'Discusses the influential teachings of mit professor gilbert strang in mathematics, particularly in the field of artificial intelligence. his lectures on linear algebra have been viewed by millions, and his book offers powerful tools for ai applications in signals, images, robotics, and quantum mechanics.', 'chapters': [{'end': 141.083, 'start': 0.069, 'title': 'Impact of mit professor on mathematics', 'summary': 'Discusses the impact of mit professor gilbert strang, known for his influential teachings in mathematics, especially in the field of artificial intelligence, with his lectures on linear algebra viewed by millions and his book providing powerful tools for ai applications, including signals, images, robotics, and quantum mechanics.', 'duration': 141.014, 'highlights': ["MIT OpenCourseWare lectures on linear algebra viewed by millions, impacting the field of artificial intelligence Gilbert Strang's lectures on linear algebra have been viewed millions of times, impacting the field of artificial intelligence and providing powerful tools for AI applications.", 'Influence on various AI applications, from signals to quantum mechanics His teachings and book have influenced various AI applications, including signals, images, robotics, computer vision, deep learning, computer graphics, and even quantum mechanics.', "Impact on education and hiring through ZipRecruiter sponsorship The podcast is supported by ZipRecruiter, emphasizing the importance of hiring great people and its effectiveness for businesses of all sizes, exemplified by Codable co-founder Gretchen Huebner's successful hiring experience."]}], 'duration': 141.014, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk069.jpg', 'highlights': ["Gilbert Strang's lectures on linear algebra have been viewed millions of times, impacting the field of artificial intelligence and providing powerful tools for AI applications.", 'His teachings and book have influenced various AI applications, including signals, images, robotics, computer vision, deep learning, computer graphics, and even quantum mechanics.', "The podcast is supported by ZipRecruiter, emphasizing the importance of hiring great people and its effectiveness for businesses of all sizes, exemplified by Codable co-founder Gretchen Huebner's successful hiring experience."]}, {'end': 1079.539, 'segs': [{'end': 187.063, 'src': 'embed', 'start': 142.298, 'weight': 0, 'content': [{'end': 146.36, 'text': 'This show is presented by Cash App, the number one finance app in the App Store.', 'start': 142.298, 'duration': 4.062}, {'end': 152.583, 'text': 'I personally use Cash App to send money to friends, but you can also use it to buy, sell, and deposit Bitcoin.', 'start': 146.96, 'duration': 5.623}, {'end': 156.686, 'text': 'Most Bitcoin exchanges take days for a bank transfer to become investable.', 'start': 153.124, 'duration': 3.562}, {'end': 158.687, 'text': 'Through Cash App, it takes seconds.', 'start': 157.126, 'duration': 1.561}, {'end': 162.489, 'text': 'Cash App also has a new investing feature.', 'start': 159.547, 'duration': 2.942}, {'end': 166.791, 'text': 'You can buy fractions of a stock, which to me is a really interesting concept.', 'start': 162.869, 'duration': 3.922}, {'end': 170.613, 'text': 'So you can buy $1 worth, no matter what the stock price is.', 'start': 167.351, 'duration': 3.262}, {'end': 177.445, 'text': 'Brokerage services are provided by Cash App Investing, a subsidiary of Square and member SIPC.', 'start': 171.616, 'duration': 5.829}, {'end': 187.063, 'text': "I'm excited to be working with Cash App to support one of my favorite organizations that many of you may know and have benefited from called FIRST,", 'start': 178.339, 'duration': 8.724}], 'summary': 'Cash app offers fast bitcoin transactions and fractional stock buying, with investing feature and partnership with first.', 'duration': 44.765, 'max_score': 142.298, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk0142298.jpg'}, {'end': 224.476, 'src': 'embed', 'start': 203.951, 'weight': 2, 'content': [{'end': 213.433, 'text': "When you get Cash App from the App Store or Google Play and use code LEXPODCAST, you'll get $10, and Cash App will also donate $10 to FIRST, which,", 'start': 203.951, 'duration': 9.482}, {'end': 220.515, 'text': "again, is an organization that I've personally seen inspire girls and boys to dream of engineering a better world.", 'start': 213.433, 'duration': 7.082}, {'end': 224.476, 'text': "And now, here's my conversation with Gilbert Strang.", 'start': 221.035, 'duration': 3.441}], 'summary': 'Use code lexpodcast on cash app for $10 and $10 donation to first', 'duration': 20.525, 'max_score': 203.951, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk0203951.jpg'}, {'end': 245.358, 'src': 'heatmap', 'start': 203.951, 'weight': 1, 'content': [{'end': 213.433, 'text': "When you get Cash App from the App Store or Google Play and use code LEXPODCAST, you'll get $10, and Cash App will also donate $10 to FIRST, which,", 'start': 203.951, 'duration': 9.482}, {'end': 220.515, 'text': "again, is an organization that I've personally seen inspire girls and boys to dream of engineering a better world.", 'start': 213.433, 'duration': 7.082}, {'end': 224.476, 'text': "And now, here's my conversation with Gilbert Strang.", 'start': 221.035, 'duration': 3.441}, {'end': 233.832, 'text': "How does it feel to be one of the modern-day rock stars of mathematics? I don't feel like a rock star.", 'start': 225.848, 'duration': 7.984}, {'end': 237.234, 'text': "That's kind of crazy for an old math person.", 'start': 233.872, 'duration': 3.362}, {'end': 245.358, 'text': "But it's true that the videos in linear algebra that I made way back in 2000,", 'start': 237.354, 'duration': 8.004}], 'summary': 'Use code lexpodcast on cash app for $10 bonus and $10 donation to first organization.', 'duration': 41.407, 'max_score': 203.951, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk0203951.jpg'}], 'start': 142.298, 'title': 'Cash app features and first sponsorship', 'summary': "Discusses cash app's features including instant bitcoin transfer, fractional stock purchasing, and sponsorship of first charity, offering a $10 donation for new users using code lexpodcast.", 'chapters': [{'end': 224.476, 'start': 142.298, 'title': 'Cash app sponsorship', 'summary': "Discusses cash app's features including instant bitcoin transfer, fractional stock purchasing, and sponsorship of first, a charity with a perfect rating on charity navigator, and offers a $10 donation to first for new users using code lexpodcast.", 'duration': 82.178, 'highlights': ['Cash App allows instant Bitcoin transfer, unlike most exchanges which take days for bank transfer to become investable.', 'Users can buy fractions of a stock, regardless of the stock price, through Cash App, offering a unique and accessible investment concept.', 'Cash App sponsors FIRST, a charity with a perfect rating on Charity Navigator, and offers a $10 donation to FIRST for new users using code LEXPODCAST.']}, {'end': 786.651, 'start': 225.848, 'title': 'Importance of linear algebra and mit opencourseware', 'summary': 'Discusses the surge in importance of linear algebra, the impact of mit opencourseware in making educational knowledge freely available, and the concept of four fundamental subspaces in linear algebra, conveyed through engaging lectures and the beauty of mathematics in high dimensions.', 'duration': 560.803, 'highlights': ['The surge in importance of linear algebra as a subject, with the videos in linear algebra being widely watched and the subject gaining recognition in recent years. The videos in linear algebra made in 2000 have been watched by millions, indicating the surge in importance of the subject in recent years.', "The impact of MIT OpenCourseWare in making educational knowledge freely available, with the decision to give away educational content being a significant and successful innovation. MIT OpenCourseWare's innovative decision to make educational content freely available, with millions of people benefiting from it, demonstrates its significant impact in the education sector.", 'The concept of four fundamental subspaces in linear algebra, including the column space, row space, and two perpendicular spaces, presented through engaging lectures and the beauty of mathematics in high dimensions. The chapter delves into the concept of four fundamental subspaces in linear algebra, emphasizing engaging lectures and the beauty of mathematics in high dimensions while conveying the concept.']}, {'end': 1079.539, 'start': 786.671, 'title': 'Linear algebra: key concepts and relevance', 'summary': 'Discusses the importance of linear algebra, emphasizing its simplicity and applicability in understanding complex multidimensional spaces, and highlights the significance of singular values and eigenvalues in breaking down matrix data for data science.', 'duration': 292.868, 'highlights': ["Linear algebra's simplicity and applicability in understanding complex multidimensional spaces, especially its flat nature and the ease of working with 10 dimensions, as compared to the complexities of calculus.", 'The significance of singular values in breaking down matrices into simple pieces, where the first and second pieces are the most important for data analysis, while the rest typically represents error or experimental noise.', 'The importance of eigenvalues in understanding square matrices and the need to extend this concept to rectangular matrices for practical applications in data science and engineering.']}], 'duration': 937.241, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk0142298.jpg', 'highlights': ['Cash App allows instant Bitcoin transfer, unlike most exchanges which take days for bank transfer to become investable.', 'Users can buy fractions of a stock, regardless of the stock price, through Cash App, offering a unique and accessible investment concept.', 'Cash App sponsors FIRST, a charity with a perfect rating on Charity Navigator, and offers a $10 donation to FIRST for new users using code LEXPODCAST.']}, {'end': 1494.629, 'segs': [{'end': 1106.68, 'src': 'embed', 'start': 1079.719, 'weight': 0, 'content': [{'end': 1085.28, 'text': "Yeah So what do you find beautiful about singular values? Well, yeah, I didn't give the theorem.", 'start': 1079.719, 'duration': 5.561}, {'end': 1089.001, 'text': "So here's the idea of singular values.", 'start': 1086.28, 'duration': 2.721}, {'end': 1099.872, 'text': 'Every matrix, every matrix, rectangular, square, whatever, can be written as a product of three very simple special matrices.', 'start': 1089.541, 'duration': 10.331}, {'end': 1100.833, 'text': "So that's the theorem.", 'start': 1099.972, 'duration': 0.861}, {'end': 1106.68, 'text': 'Every matrix can be written as a rotation times a stretch, which is a..', 'start': 1101.414, 'duration': 5.266}], 'summary': 'Every matrix can be written as a product of three special matrices.', 'duration': 26.961, 'max_score': 1079.719, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01079719.jpg'}, {'end': 1197.453, 'src': 'embed', 'start': 1162.328, 'weight': 1, 'content': [{'end': 1164.909, 'text': "Up to 10 dimensions, you've got 10 ways to turn.", 'start': 1162.328, 'duration': 2.581}, {'end': 1167.99, 'text': 'But you can visualize a rotation.', 'start': 1165.429, 'duration': 2.561}, {'end': 1169.771, 'text': 'Take the space and turn it.', 'start': 1168.53, 'duration': 1.241}, {'end': 1172.011, 'text': 'And you can visualize a stretch.', 'start': 1170.431, 'duration': 1.58}, {'end': 1173.112, 'text': 'So to break a..', 'start': 1172.091, 'duration': 1.021}, {'end': 1183.323, 'text': 'a matrix with all those numbers in it into something you can visualize, rotate, stretch, rotate.', 'start': 1174.778, 'duration': 8.545}, {'end': 1184.484, 'text': "It's pretty neat.", 'start': 1183.704, 'duration': 0.78}, {'end': 1185.585, 'text': "It's pretty neat.", 'start': 1184.844, 'duration': 0.741}, {'end': 1186.525, 'text': "That's pretty powerful.", 'start': 1185.705, 'duration': 0.82}, {'end': 1197.453, 'text': 'On YouTube, just consuming a bunch of videos and just watching what people connect with and what they really enjoy and are inspired by,', 'start': 1187.686, 'duration': 9.767}], 'summary': '10 dimensions offer 10 ways to rotate and stretch, making visualization powerful.', 'duration': 35.125, 'max_score': 1162.328, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01162328.jpg'}, {'end': 1267.953, 'src': 'embed', 'start': 1236.537, 'weight': 3, 'content': [{'end': 1246.34, 'text': "A lot of people quite like math, and I get messages from people saying, you know, now I'm retired, I'm gonna learn some more math.", 'start': 1236.537, 'duration': 9.803}, {'end': 1249.482, 'text': "I get a lot of those, it's really encouraging.", 'start': 1246.86, 'duration': 2.622}, {'end': 1259.692, 'text': "And I think what people like is that there's some order, a lot of order, and things are not obvious, but they're true.", 'start': 1250.003, 'duration': 9.689}, {'end': 1267.953, 'text': "So it's really cheering to think that so many people really want to learn more about math.", 'start': 1260.572, 'duration': 7.381}], 'summary': 'Many retirees express interest in learning math, finding order and truth in it.', 'duration': 31.416, 'max_score': 1236.537, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01236537.jpg'}, {'end': 1359.874, 'src': 'embed', 'start': 1327.387, 'weight': 4, 'content': [{'end': 1330.388, 'text': 'Or counting, yeah.', 'start': 1327.387, 'duration': 3.001}, {'end': 1334.457, 'text': 'That was a source of peace almost.', 'start': 1331.916, 'duration': 2.541}, {'end': 1342.358, 'text': 'What is it about math do you think that brings that? What is that? Well, you know where you are.', 'start': 1336.677, 'duration': 5.681}, {'end': 1343.779, 'text': "It's symmetry.", 'start': 1342.378, 'duration': 1.401}, {'end': 1345.259, 'text': "It's certainty.", 'start': 1343.899, 'duration': 1.36}, {'end': 1352.881, 'text': 'The fact that if you multiply two by itself 10 times, you get 1,024, period.', 'start': 1345.859, 'duration': 7.022}, {'end': 1354.461, 'text': "Everybody's going to get that.", 'start': 1353.341, 'duration': 1.12}, {'end': 1359.874, 'text': "Do you see math as a powerful tool or as an art form? So it's both.", 'start': 1354.961, 'duration': 4.913}], 'summary': 'Math provides peace through symmetry and certainty, exemplified by the universal understanding of multiplying 2 by itself 10 times resulting in 1,024.', 'duration': 32.487, 'max_score': 1327.387, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01327387.jpg'}, {'end': 1429.032, 'src': 'embed', 'start': 1393.44, 'weight': 5, 'content': [{'end': 1406.914, 'text': 'And yeah, so probably within the MIT math department, most people enjoy teaching students who get the abstract idea.', 'start': 1393.44, 'duration': 13.474}, {'end': 1415.002, 'text': "I'm okay with, I'm good with engineers who are looking for a way to find answers, yeah.", 'start': 1407.074, 'duration': 7.928}, {'end': 1417.684, 'text': "Actually, that's an interesting question.", 'start': 1415.542, 'duration': 2.142}, {'end': 1429.032, 'text': "Do you think for teaching and in general thinking about new concepts, do you think it's better to plug in the numbers or to think more abstractly?", 'start': 1417.704, 'duration': 11.328}], 'summary': 'In the mit math department, most prefer teaching abstract concepts. some are comfortable with engineers seeking concrete answers.', 'duration': 35.592, 'max_score': 1393.44, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01393440.jpg'}], 'start': 1079.719, 'title': 'Matrix decomposition and math in youtube culture', 'summary': 'Discusses the theorem of matrix decomposition and its visual appeal, along with the popularity of math on youtube, its wide audience appeal, and the debate between abstract thinking and practical application.', 'chapters': [{'end': 1186.525, 'start': 1079.719, 'title': 'Singular values and matrix decomposition', 'summary': 'Discusses the theorem stating that every matrix can be written as a product of three simple special matrices, which are rotation, stretch, and another rotation, making it visually appealing and powerful.', 'duration': 106.806, 'highlights': ['The theorem states that every matrix can be written as a rotation times a stretch, which is a diagonal matrix, and then the third factor is another rotation, making it visually appealing and powerful.', 'Geometrically, the ability to break a matrix into a rotation, stretch, and another rotation is visually appealing and powerful, as it allows visualization in up to 10 dimensions and makes the structure of the matrix neat and powerful.', 'The discussion highlights the visualization of rotations and stretches in two-dimensional and three-dimensional spaces, emphasizing their significance in understanding the decomposition of matrices.']}, {'end': 1494.629, 'start': 1187.686, 'title': 'The appeal of math in youtube culture', 'summary': 'Explores the popularity of math on youtube, highlighting its appeal to a wide audience, including retirees, and discusses the comforting nature of math, its role as both a powerful tool and an art form, and the debate between abstract thinking and practical application.', 'duration': 306.943, 'highlights': ["Math's appeal on YouTube, including retirees seeking to learn more math, is discussed The speaker receives messages from retirees expressing their desire to learn more math, indicating the wide appeal of math on YouTube.", 'Discussion on the comforting nature of math, its role as a powerful tool, and an art form The conversation delves into the comforting and orderly nature of math, its role as both a powerful tool and an art form, reflecting its versatility and appeal.', 'Debate between abstract thinking and practical application in math The debate between abstract thinking and practical application in math is explored, with a focus on the approach to teaching and understanding new concepts.']}], 'duration': 414.91, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01079719.jpg', 'highlights': ['The theorem states that every matrix can be written as a rotation times a stretch, which is a diagonal matrix, and then the third factor is another rotation, making it visually appealing and powerful.', 'Geometrically, the ability to break a matrix into a rotation, stretch, and another rotation is visually appealing and powerful, as it allows visualization in up to 10 dimensions and makes the structure of the matrix neat and powerful.', 'The discussion highlights the visualization of rotations and stretches in two-dimensional and three-dimensional spaces, emphasizing their significance in understanding the decomposition of matrices.', "Math's appeal on YouTube, including retirees seeking to learn more math, is discussed The speaker receives messages from retirees expressing their desire to learn more math, indicating the wide appeal of math on YouTube.", 'Discussion on the comforting nature of math, its role as a powerful tool, and an art form The conversation delves into the comforting and orderly nature of math, its role as both a powerful tool and an art form, reflecting its versatility and appeal.', 'Debate between abstract thinking and practical application in math The debate between abstract thinking and practical application in math is explored, with a focus on the approach to teaching and understanding new concepts.']}, {'end': 1983.801, 'segs': [{'end': 1525.229, 'src': 'embed', 'start': 1494.669, 'weight': 0, 'content': [{'end': 1497.351, 'text': 'Right, but nevertheless working in the space of examples.', 'start': 1494.669, 'duration': 2.682}, {'end': 1498.291, 'text': 'Yeah, examples.', 'start': 1497.511, 'duration': 0.78}, {'end': 1499.152, 'text': 'It seems to.', 'start': 1498.311, 'duration': 0.841}, {'end': 1501.473, 'text': 'Examples of structure.', 'start': 1499.272, 'duration': 2.201}, {'end': 1503.374, 'text': 'Our brains seem to connect with that.', 'start': 1501.873, 'duration': 1.501}, {'end': 1504.415, 'text': 'Yeah, yeah.', 'start': 1503.674, 'duration': 0.741}, {'end': 1510.038, 'text': "So I'm not sure if you're familiar with him, but Andrew Yang is a presidential candidate.", 'start': 1504.655, 'duration': 5.383}, {'end': 1518.683, 'text': 'currently running with a math in all capital letters and his hats as a slogan.', 'start': 1511.857, 'duration': 6.826}, {'end': 1519.103, 'text': 'I see.', 'start': 1518.943, 'duration': 0.16}, {'end': 1521.265, 'text': 'It stands for Make America Think Hard.', 'start': 1519.223, 'duration': 2.042}, {'end': 1525.229, 'text': "okay, i'll vote for him.", 'start': 1521.807, 'duration': 3.422}], 'summary': "Discussion on andrew yang's presidential campaign and his slogan 'make america think hard.'", 'duration': 30.56, 'max_score': 1494.669, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01494669.jpg'}, {'end': 1631.016, 'src': 'embed', 'start': 1548.645, 'weight': 1, 'content': [{'end': 1550.947, 'text': 'well, what are your thoughts on that in general?', 'start': 1548.645, 'duration': 2.302}, {'end': 1558.218, 'text': 'Well, certainly somewhere in the system we need people who are comfortable with numbers, comfortable with quantities.', 'start': 1551.175, 'duration': 7.043}, {'end': 1565.622, 'text': "If you say this leads to that, they see it and it's undeniable.", 'start': 1559.519, 'duration': 6.103}, {'end': 1569.804, 'text': "But isn't that strange to you that we have almost no,", 'start': 1565.982, 'duration': 3.822}, {'end': 1583.313, 'text': "I mean I'm pretty sure we have no elected officials in Congress or obviously the president that either has an engineering degree or a math degree.", 'start': 1569.804, 'duration': 13.509}, {'end': 1584.875, 'text': "Yeah, well, that's too bad.", 'start': 1583.354, 'duration': 1.521}, {'end': 1591.401, 'text': 'A few who could make the connection.', 'start': 1585.896, 'duration': 5.505}, {'end': 1603.936, 'text': 'it would have to be people who understand engineering or science and, at the same time, can make speeches and lead yeah.', 'start': 1591.401, 'duration': 12.535}, {'end': 1605.457, 'text': 'Yeah, inspire people.', 'start': 1604.416, 'duration': 1.041}, {'end': 1606.578, 'text': 'Yeah, inspire, yeah.', 'start': 1605.537, 'duration': 1.041}, {'end': 1612.422, 'text': 'You were, speaking of inspiration, the president of the Society for Industrial and Applied Mathematics.', 'start': 1606.618, 'duration': 5.804}, {'end': 1613.243, 'text': 'Oh, yes.', 'start': 1612.963, 'duration': 0.28}, {'end': 1616.906, 'text': "It's a major organization in math, in applied math.", 'start': 1613.263, 'duration': 3.643}, {'end': 1623.251, 'text': 'What do you see as the role of that society, you know, in our public discourse? Right.', 'start': 1618.127, 'duration': 5.124}, {'end': 1627.434, 'text': 'Yeah, so, well, it was fun to be president at the time.', 'start': 1623.791, 'duration': 3.643}, {'end': 1628.355, 'text': 'at the time.', 'start': 1627.794, 'duration': 0.561}, {'end': 1629.415, 'text': 'A couple years, a few years.', 'start': 1628.375, 'duration': 1.04}, {'end': 1631.016, 'text': 'Two years, around 2000.', 'start': 1629.435, 'duration': 1.581}], 'summary': 'The lack of elected officials with engineering or math degrees is noted, and the importance of merging science and leadership is emphasized.', 'duration': 82.371, 'max_score': 1548.645, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01548645.jpg'}, {'end': 1891.165, 'src': 'embed', 'start': 1828.405, 'weight': 3, 'content': [{'end': 1840.634, 'text': 'And this idea of a neural net is part of the structure of our new way to connect create a rule.', 'start': 1828.405, 'duration': 12.229}, {'end': 1848.077, 'text': "We're looking for a rule that will take these training inputs to the known outputs.", 'start': 1840.674, 'duration': 7.403}, {'end': 1855.501, 'text': "And then we're gonna use that rule on new inputs that we don't know the output and see what comes.", 'start': 1848.437, 'duration': 7.064}, {'end': 1859.002, 'text': 'Linear algebra is a big part of finding that rule.', 'start': 1856.141, 'duration': 2.861}, {'end': 1859.802, 'text': "That's right.", 'start': 1859.222, 'duration': 0.58}, {'end': 1861.523, 'text': 'Linear algebra is a big part.', 'start': 1860.023, 'duration': 1.5}, {'end': 1862.764, 'text': 'Not all the part.', 'start': 1861.943, 'duration': 0.821}, {'end': 1866.046, 'text': 'People were leaning on matrices.', 'start': 1863.504, 'duration': 2.542}, {'end': 1866.726, 'text': "That's good.", 'start': 1866.126, 'duration': 0.6}, {'end': 1867.367, 'text': 'Still do.', 'start': 1866.787, 'duration': 0.58}, {'end': 1870.089, 'text': 'Linear is something special.', 'start': 1868.308, 'duration': 1.781}, {'end': 1873.251, 'text': "It's all about straight lines and flat planes.", 'start': 1870.349, 'duration': 2.902}, {'end': 1878.675, 'text': "And data isn't quite like that.", 'start': 1874.172, 'duration': 4.503}, {'end': 1881.157, 'text': "It's more complicated.", 'start': 1878.896, 'duration': 2.261}, {'end': 1883.139, 'text': 'So you got to introduce some complication.', 'start': 1881.217, 'duration': 1.922}, {'end': 1887.162, 'text': "So you have to have some function that's not a straight line.", 'start': 1883.719, 'duration': 3.443}, {'end': 1891.165, 'text': 'And it turned out nonlinear, nonlinear, nonlinear.', 'start': 1887.462, 'duration': 3.703}], 'summary': 'Using linear algebra and neural nets to find rules for complex data processing.', 'duration': 62.76, 'max_score': 1828.405, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01828405.jpg'}], 'start': 1494.669, 'title': "Math's role in politics", 'summary': "Discusses the link between math and politics using andrew yang's campaign as an example, and emphasizes the role of math and engineering in public leadership, including the influence of the society for industrial and applied mathematics and the significance of deep learning in understanding data and ai.", 'chapters': [{'end': 1569.804, 'start': 1494.669, 'title': 'Math and politics connection', 'summary': "Discusses the connection between math and politics, highlighting the example of andrew yang's presidential campaign and the lack of emphasis on math, science, and engineering in political discourse.", 'duration': 75.135, 'highlights': ["The example of Andrew Yang's presidential campaign with a math-focused slogan 'Make America Think Hard' is discussed, highlighting the connection between math and politics.", 'The lack of emphasis on math, science, and engineering in political discourse and government is noted as a notable observation.', 'The importance of having individuals comfortable with numbers and quantities in the political system is highlighted as a key point for consideration.']}, {'end': 1983.801, 'start': 1569.804, 'title': 'Role of math in public discourse', 'summary': 'Discusses the role of math and engineering in public leadership, the influence of the society for industrial and applied mathematics, and the significance of deep learning in understanding data and artificial intelligence.', 'duration': 413.997, 'highlights': ['The significance of math and engineering in public leadership and the need for individuals with expertise in these fields who can inspire and lead.', "The president's role in the Society for Industrial and Applied Mathematics, its impact on public discourse, and the attention math received in Washington during the early 2000s.", 'The relevance of deep learning in understanding vast amounts of data and its use of linear algebra to create rules for new inputs and outputs.', 'The explanation of piecewise linear functions and their role in making neural networks sufficiently complicated and realistic in understanding complex data.']}], 'duration': 489.132, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01494669.jpg', 'highlights': ["The example of Andrew Yang's presidential campaign with a math-focused slogan 'Make America Think Hard' is discussed, highlighting the connection between math and politics.", 'The importance of having individuals comfortable with numbers and quantities in the political system is highlighted as a key point for consideration.', 'The significance of math and engineering in public leadership and the need for individuals with expertise in these fields who can inspire and lead.', 'The relevance of deep learning in understanding vast amounts of data and its use of linear algebra to create rules for new inputs and outputs.', 'The explanation of piecewise linear functions and their role in making neural networks sufficiently complicated and realistic in understanding complex data.', 'The lack of emphasis on math, science, and engineering in political discourse and government is noted as a notable observation.', "The president's role in the Society for Industrial and Applied Mathematics, its impact on public discourse, and the attention math received in Washington during the early 2000s."]}, {'end': 2244.068, 'segs': [{'end': 2014.6, 'src': 'embed', 'start': 1984.352, 'weight': 1, 'content': [{'end': 1992.176, 'text': 'This idea of things that are piecewise, linear, flat pieces but with folds between them,', 'start': 1984.352, 'duration': 7.824}, {'end': 2002.882, 'text': 'like think of a roof of an infinitely complicated house or something that almost curved, but every piece is flat.', 'start': 1992.176, 'duration': 10.706}, {'end': 2006.765, 'text': "That's been used by engineers.", 'start': 2004.743, 'duration': 2.022}, {'end': 2008.766, 'text': 'That idea has been used by engineers.', 'start': 2006.805, 'duration': 1.961}, {'end': 2014.6, 'text': 'is used by engineers big time, something called the finite element method.', 'start': 2009.939, 'duration': 4.661}], 'summary': 'Engineers use the finite element method for piecewise flat structures.', 'duration': 30.248, 'max_score': 1984.352, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01984352.jpg'}, {'end': 2109.271, 'src': 'embed', 'start': 2073.783, 'weight': 0, 'content': [{'end': 2075.603, 'text': "A supercomputer isn't going to do it.", 'start': 2073.783, 'duration': 1.82}, {'end': 2079.344, 'text': 'But those keep getting more powerful.', 'start': 2076.083, 'duration': 3.261}, {'end': 2087.485, 'text': 'So that limit has been moved to allow more and more complicated surfaces.', 'start': 2079.484, 'duration': 8.001}, {'end': 2095.228, 'text': 'So in terms of just mapping from inputs to outputs, looking at data, what do you think..', 'start': 2087.525, 'duration': 7.703}, {'end': 2103.949, 'text': 'in the context of neural networks in general, data is just tensor vectors, matrices, tensors.', 'start': 2097.966, 'duration': 5.983}, {'end': 2109.271, 'text': 'How do you think about learning from data??', 'start': 2105.99, 'duration': 3.281}], 'summary': 'Advancements in supercomputers allow for more complex data mapping and learning from data using neural networks.', 'duration': 35.488, 'max_score': 2073.783, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk02073783.jpg'}, {'end': 2244.068, 'src': 'embed', 'start': 2149.039, 'weight': 3, 'content': [{'end': 2166.41, 'text': 'Newton and Einstein and other great, great people have found those laws and laws of the distribution of oil in a, in an underground thing.', 'start': 2149.039, 'duration': 17.371}, {'end': 2176.997, 'text': 'So petroleum engineers understand how oil will sit in an underground basin.', 'start': 2166.671, 'duration': 10.326}, {'end': 2179.839, 'text': 'So there were rules.', 'start': 2178.218, 'duration': 1.621}, {'end': 2186.483, 'text': 'Now the new idea of artificial intelligence is Learn the rules.', 'start': 2180.099, 'duration': 6.384}, {'end': 2195.206, 'text': 'Instead of figuring out the rules with help from Newton or Einstein, the computer is looking for the rules.', 'start': 2186.923, 'duration': 8.283}, {'end': 2196.766, 'text': "So that's another step.", 'start': 2195.646, 'duration': 1.12}, {'end': 2205.208, 'text': "But if there are no rules at all that the computer could find, if it's totally random data, well, you've got nothing.", 'start': 2196.886, 'duration': 8.322}, {'end': 2207.889, 'text': "You've got no science to discover.", 'start': 2205.348, 'duration': 2.541}, {'end': 2211.096, 'text': "It's an automated search for the underlying rules.", 'start': 2208.334, 'duration': 2.762}, {'end': 2213.318, 'text': 'Yeah, search for the rules.', 'start': 2211.416, 'duration': 1.902}, {'end': 2214.399, 'text': 'Yeah, exactly.', 'start': 2213.398, 'duration': 1.001}, {'end': 2217.721, 'text': 'And there will be a lot of random parts.', 'start': 2214.859, 'duration': 2.862}, {'end': 2222.465, 'text': "I'm not knocking random because that's there.", 'start': 2218.522, 'duration': 3.943}, {'end': 2229.29, 'text': "There's a lot of randomness built in, but there's got to be some basic structure.", 'start': 2223.025, 'duration': 6.265}, {'end': 2232.753, 'text': "It's almost always signal, right? There's got to be some signal, yeah.", 'start': 2229.31, 'duration': 3.443}, {'end': 2237.016, 'text': "If it's all noise, then you're not going to get anywhere.", 'start': 2232.793, 'duration': 4.223}, {'end': 2244.068, 'text': 'Well, this world around us does seem to always have a signal of some kind to be discovered.', 'start': 2237.421, 'duration': 6.647}], 'summary': 'Ai seeks underlying rules in data, aiming to find structure and signal amidst randomness.', 'duration': 95.029, 'max_score': 2149.039, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk02149039.jpg'}], 'start': 1984.352, 'title': "Expressive power and ai's search for rules", 'summary': "Explores expressive power of piecewise flat functions in engineering applications and discusses ai's search for underlying rules in data, emphasizing the need for structure amidst randomness.", 'chapters': [{'end': 2125.819, 'start': 1984.352, 'title': 'Expressive power of piecewise flat functions', 'summary': 'Explores the expressive power of piecewise flat functions, particularly in the context of engineering applications such as the finite element method, and discusses the increasing computational capabilities of neural networks in mapping complex surfaces and learning from data.', 'duration': 141.467, 'highlights': ['The finite element method is extensively used by engineers in designing bridges, buildings, and airplanes, leveraging the expressive power of piecewise flat as a simple computable approximation.', 'The computational limits of neural networks in mapping complex surfaces have been continuously expanded with the increasing power of supercomputers.', 'Deep learning revolves around the idea of learning from data and exploring the expressive capabilities of neural networks in mapping and understanding the complexities of real-world data.']}, {'end': 2244.068, 'start': 2126.299, 'title': "Ai's search for rules", 'summary': 'Discusses the importance of finding underlying rules in data for artificial intelligence, highlighting the significance of rules in various fields and emphasizing the need for a basic structure amidst randomness.', 'duration': 117.769, 'highlights': ['The importance of finding underlying rules in data for artificial intelligence is emphasized, showcasing the significance of rules in various fields such as physics and petroleum engineering.', 'The new idea of artificial intelligence involves the computer learning the rules rather than relying on human assistance, marking a significant step forward.', 'The presence of randomness in data is acknowledged, but it is emphasized that there must be some basic structure and signal for meaningful discovery to occur.']}], 'duration': 259.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk01984352.jpg', 'highlights': ['The computational limits of neural networks in mapping complex surfaces have been continuously expanded with the increasing power of supercomputers.', 'The finite element method is extensively used by engineers in designing bridges, buildings, and airplanes, leveraging the expressive power of piecewise flat as a simple computable approximation.', 'Deep learning revolves around the idea of learning from data and exploring the expressive capabilities of neural networks in mapping and understanding the complexities of real-world data.', 'The importance of finding underlying rules in data for artificial intelligence is emphasized, showcasing the significance of rules in various fields such as physics and petroleum engineering.', 'The new idea of artificial intelligence involves the computer learning the rules rather than relying on human assistance, marking a significant step forward.', 'The presence of randomness in data is acknowledged, but it is emphasized that there must be some basic structure and signal for meaningful discovery to occur.']}, {'end': 2984.004, 'segs': [{'end': 2509.871, 'src': 'embed', 'start': 2443.5, 'weight': 0, 'content': [{'end': 2454.647, 'text': "very useful that's what's happened in my lifetime is the importance of data, which does come in matrix form.", 'start': 2443.5, 'duration': 11.147}, {'end': 2458.61, 'text': "so it's really set up for algebra, it's not set up for differential equations.", 'start': 2454.647, 'duration': 3.963}, {'end': 2470.598, 'text': 'And let me fairly add, probability, the ideas of probability and statistics have become very, very important, have also jumped forward.', 'start': 2459.29, 'duration': 11.308}, {'end': 2474.682, 'text': "And that's different from linear algebra, quite different.", 'start': 2471.739, 'duration': 2.943}, {'end': 2488.396, 'text': 'So now we really have three major areas to me, calculus, linear algebra, matrices, and probability statistics.', 'start': 2475.222, 'duration': 13.174}, {'end': 2493.462, 'text': 'And they all deserve an important place.', 'start': 2489.237, 'duration': 4.225}, {'end': 2500.586, 'text': "And calculus has traditionally had a lion's share of the time.", 'start': 2494.062, 'duration': 6.524}, {'end': 2501.846, 'text': 'A disproportionate share.', 'start': 2500.666, 'duration': 1.18}, {'end': 2502.767, 'text': 'It is, thank you.', 'start': 2501.866, 'duration': 0.901}, {'end': 2504.488, 'text': "Disproportionate, that's a good word.", 'start': 2502.827, 'duration': 1.661}, {'end': 2509.871, 'text': 'Of the love and attention from the excited young minds.', 'start': 2505.789, 'duration': 4.082}], 'summary': 'Data importance in lifetime: focus on algebra, probability, statistics, and calculus.', 'duration': 66.371, 'max_score': 2443.5, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk02443500.jpg'}, {'end': 2624.49, 'src': 'embed', 'start': 2597.446, 'weight': 2, 'content': [{'end': 2610.337, 'text': 'Second, my matrices, which are like the linear algebra version of second derivatives, are neat in linear algebra, yeah.', 'start': 2597.446, 'duration': 12.891}, {'end': 2612.599, 'text': 'Just everything comes out right with those guys.', 'start': 2610.477, 'duration': 2.122}, {'end': 2614.381, 'text': 'Yeah Beautiful.', 'start': 2613.941, 'duration': 0.44}, {'end': 2624.49, 'text': 'What did you learn about the process of learning by having taught so many students math over the years? Ooh, that is hard.', 'start': 2615.422, 'duration': 9.068}], 'summary': 'Matrices in linear algebra are neat, making everything come out right.', 'duration': 27.044, 'max_score': 2597.446, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk02597446.jpg'}, {'end': 2729.046, 'src': 'embed', 'start': 2701.886, 'weight': 3, 'content': [{'end': 2706.37, 'text': "Are there moments of learning that you just see in the student's eyes?", 'start': 2701.886, 'duration': 4.484}, {'end': 2711.575, 'text': "You don't need to look at the grades, but you see in their eyes that you hook them,", 'start': 2706.39, 'duration': 5.185}, {'end': 2721.223, 'text': 'that you connect with them in a way where you know what they fall in love with this, with this beautiful world of math.', 'start': 2711.575, 'duration': 9.648}, {'end': 2724.324, 'text': "They see that it's got some beauty there.", 'start': 2721.323, 'duration': 3.001}, {'end': 2727.846, 'text': 'Or conversely, that they give up at that point.', 'start': 2724.544, 'duration': 3.302}, {'end': 2729.046, 'text': "It's the opposite.", 'start': 2728.166, 'duration': 0.88}], 'summary': "Recognizing students' love for math beyond just grades is essential for impactful teaching.", 'duration': 27.16, 'max_score': 2701.886, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk02701886.jpg'}], 'start': 2244.308, 'title': "Linear algebra's significance", 'summary': "Discusses the importance of linear algebra in understanding high-dimensional surfaces and handling data in matrix form, distinguishing it from calculus and probability statistics. it also explores gilbert strang's passionate teaching approach and its positive impact on students, focusing on the joy of teaching and the beauty in mathematics.", 'chapters': [{'end': 2596.559, 'start': 2244.308, 'title': 'Linear algebra importance', 'summary': 'Discusses the significance of linear algebra in mathematics, particularly its relevance in understanding high-dimensional surfaces and the increasing importance of data in matrix form, setting it apart from calculus and probability statistics.', 'duration': 352.251, 'highlights': ['The increasing importance of data in matrix form makes linear algebra very useful, particularly in high-dimensional spaces. Linear algebra is highlighted as being very useful, especially in high-dimensional spaces, due to the increasing importance of data in matrix form.', 'The disproportionate attention given to calculus compared to linear algebra in undergraduate mathematics is noted. The speaker emphasizes the disproportionate attention given to calculus compared to linear algebra in undergraduate mathematics.', "Discussion on the speaker's favorite matrix, which is a square matrix with twos running down the main diagonal and non-zero diagonals coming down, and its relevance in engineering and calculus. The speaker shares details about their favorite matrix and its relevance in engineering and calculus, particularly in understanding second derivatives and its fundamental role in understanding space."]}, {'end': 2984.004, 'start': 2597.446, 'title': 'Teaching linear algebra with passion', 'summary': "Discusses gilbert strang's approach to teaching linear algebra, emphasizing the joy of teaching and the impact on students, while downplaying the importance of grading and assessment, and highlighting the beauty and patterns in mathematics.", 'duration': 386.558, 'highlights': ['Gilbert Strang emphasizes the joy of teaching and the impact on students, while downplaying the importance of grading and assessment. Strang expresses his focus on teaching and connecting with students, while de-emphasizing grading and assessment, showcasing a student-centered approach to education.', 'Strang highlights the beauty and patterns in mathematics, emphasizing the joy of teaching and learning. Strang expresses his passion for teaching and the beauty of mathematics, highlighting the moments where students connect with the subject and the joy of teaching.', 'Strang discusses the impact of his teaching on students, receiving positive messages from those who have watched his videos. Strang reflects on the impact of his teaching, mentioning the positive feedback received from students who have benefitted from his videos, showcasing the reach and influence of his teaching.']}], 'duration': 739.696, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/lEZPfmGCEk0/pics/lEZPfmGCEk02244308.jpg', 'highlights': ["Linear algebra's usefulness in high-dimensional spaces due to the increasing importance of data in matrix form.", 'Disproportionate attention given to calculus compared to linear algebra in undergraduate mathematics.', "Favorite matrix's relevance in engineering and calculus, particularly in understanding second derivatives and space.", "Gilbert Strang's emphasis on the joy of teaching and the impact on students, while downplaying the importance of grading and assessment.", "Strang's focus on teaching and connecting with students, while de-emphasizing grading and assessment, showcasing a student-centered approach to education.", "Strang's passion for teaching and the beauty of mathematics, highlighting the moments where students connect with the subject and the joy of teaching."]}], 'highlights': ["Gilbert Strang's lectures on linear algebra have impacted the field of artificial intelligence and various AI applications.", 'Cash App allows instant Bitcoin transfer and offers unique and accessible investment concepts.', "Matrix decomposition's visual appeal allows visualization in up to 10 dimensions and makes the structure of the matrix neat and powerful.", 'The connection between math and politics is highlighted, emphasizing the importance of individuals comfortable with numbers and quantities in the political system.', 'The computational limits of neural networks in mapping complex surfaces have been continuously expanded with the increasing power of supercomputers.', "Linear algebra's usefulness in high-dimensional spaces due to the increasing importance of data in matrix form."]}