Linear Algebra Review for Big Data Summer Institute
Join Rupam Bhattacharyya at the Big Data Summer Institute for a comprehensive review of linear algebra concepts. Explore topics such as matrix notation, special matrices, shapes of matrices, and matrix operations. Gain valuable insights for applications in big data analysis and machine learning.
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Big Data Summer Institute Big Data Summer Institute Linear Algebra Review Linear Algebra Review June 23, 2022; 1:30-2:45 pm Rupam Bhattacharyya BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 1
Housekeeping Information Housekeeping Information Instructor: Rupam Bhattacharyya PhD Candidate, Biostatistics, University of Michigan rupamb@umich.edu Useful Materials: Listed throughout the slides. Some additional slides at the end. Acknowledgements Lap Sum Chan, Holly Hartman. YT and Google. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 2
Dimensions of Objects Dimensions of Objects https://dev.to/mmithrakumar/scalars-vectors-matrices-and-tensors-with-tensorflow-2-0-1f66 BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 3
The Matrix Notation The Matrix Notation https://upload.wikimedia.org/wikipedia/commons/8/84/Matrix.png BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 4
Shapes Shapes ? < ? ? > ? ? = ? Rectangular/Non-square Square BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 5
Special Matrices Special Matrices Diagonal matrices. Restricted to square matrices for us. Identity matrices. What are identity matrices of dimensions 1, 2, ? Zero matrices. Not restricted to any shape. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 6
Special Matrices [Contd.] Special Matrices [Contd.] Upper triangular. ???= ? ?? ? > ?. Lower triangular. ???= ? ?? ? < ?. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 7
Matrix Operations: Addition Matrix Operations: Addition Basically element-wise addition. Which means if you want to add two matrices, they must have the same number of rows and same number of columns. Impossible. Possible. Commutative:? + ? = ? + ?. Associative: ? + ? + ? = ? + (? + ?). BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 8
Matrix Operations: Multiplication Matrix Operations: Multiplication https://gfycat.com/positiveexhaustedamericangoldfinch BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 9
Multiplication [Contd.] Multiplication [Contd.] Not element-wise anymore. Number of columns of the first matrix must be same as number of rows of the second. Associative: ?? ? = ?(??). Distributive:? ? + ? ? = ??? + ???. NOT commutative:?? ??. Both may not even exist. Example? Even if they do, may have different dimensions. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 10
Other Products Other Products Hadamard product: like addition in many ways. Kronecker product: block-wise operation. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 11
Matrix Operations: Transpose Matrix Operations: Transpose https://www.codecademy.com/learn/learn-linear-algebra/modules/math-ds-linear-algebra/cheatsheet Pretty obvious that this will be well-behaved with element-wise operations. ? + ??= ??+ ??. The procedure undoes itself, i.e., ???= ?. Multiplication? ???= ????. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 12
Size of a Matrix Size of a Matrix Target: Establish rules similar to scalar numbers in the matrix world. So far, we have been able to achieve addition (thus subtraction) and some version of multiplication. To bring in division, we need some idea of size since division is essentially estimating size in the scalar world. 10/2 = 5 because in our very natural sense of size, 10 is 5 times the value of 2. 3/9 < 1 because 3 is smaller in size than 9. Need to define some size function that can take a [square] matrix as an input and compute a scalar number as output. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 13
Size Function: Trace Size Function: Trace Easy to see that this works in a linear fashion. ?? ??= ??(?). ?? ? + ? = ?? ? + ??(?). What about multiplication? ?? ?? = ??(??). Not enough! BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 14
Determinant Determinant https://www.chilimath.com/lessons/advanced-algebra/determinant-2x2-matrix/ There is a general definition for higher dimension square matrices that we will skip here. https://www.chilimath.com/lessons/advanced-algebra/determinant-3x3-matrix/ BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 15
To Division and Inverses To Division and Inverses In the scalar world, division is basically a multiplication, just like subtraction is basically an addition. 4 2= 4 1 As long as we can define ? 1 from ?, we can go on to compute things like ? 1?,?? 1 and so on. As you would expect, determinant is connected to this. Just like you can t invert zero among scalars, you can t invert matrices with determinant zero. 2. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 16
Matrix Inversion Matrix Inversion Can be extended to higher order exactly like determinant. Properties? ?? 1= ? 1? = ?. ?? 1= ? 1? 1. ? 1 ?= ?? 1. 1 det ?? = det ? det ? ,det ? 1= det(?). BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 17
Extras Extras Eigenvalues and eigenvectors. Rank of a matrix + connections with det and inverse. Projection matrices and linear regression. Matrices in R. BDSI 2022 Week 1 Linear Algebra Review | Rupam Bhattacharyya 18
