3. Linear Algebra for Machine Learning: Factorization and Linear Transformations

1. Seminar Series on Linear Algebra for Machine Learning Part 3: Factorization and Linear Transformations Dr. Ceni Babaoglu Data Science Laboratory Ryerson University cenibabaoglu.com Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

2. Overview 1 Row and Column Spaces 2 Rank of a Matrix 3 Rank and Singularity 4 Inner Product Spaces 5 Gram-Schmidt Process 6 Factorization 7 Linear Transformation 8 Linear Transformation and Singularity 9 Similar Matrices 10 References Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

3. Row and Column Spaces Let A =      a11 a12 a13 . . . a1n a21 a22 a23 . . . a2n ... ... ... ... ... am1 am2 am3 . . . amn      be an m × n matrix. The rows of A, considered as vectors in Rn, span a subspace of Rn called the row space of A. Similarly, the columns of A, considered as vectors in Rm, span a subspace of Rm called the column space of A. If A and B are two m × n row (column) equivalent matrices, then the row (column) spaces of A and B are equal. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

4. Rank of a Matrix The dimension of the row (column) space of A is called the row (column) rank of A. The row rank and column rank of the m × n matrix A = [aij ] are equal. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

5. Rank and Singularity A is nonsingular. Ax = 0 has only the trivial solution. A is row (column) equivalent to In. For every vector b in Rn, the system Ax = b has a unique solution. det(A) = 0. The rank of A is n. The rows of A form a linearly independent set of vectors in Rn. The columns of A form a linearly independent set of vectors in Rn. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

6. Example Let A =   1 −1 2 0 −3 0 1 0 4 0 2 −1 4 4 −6  . Find the following: (i) A basis for the column space of A and its dimension. (ii) A basis for the row space of A and its dimension. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

7. Example   1 −1 2 0 −3 0 1 0 4 0 2 −1 4 4 −6   S3−2S1→S3 −−−−−−−→   1 −1 2 0 −3 0 1 0 4 0 0 1 0 4 0   S1+S2→S1 S3−S2→S3 −−−−−−→   1 0 2 4 −3 0 1 0 4 0 0 0 0 0 0   (i) The column space of A is spanned by the vectors (1, 0, 2)T and (−1, 1, −1)T . These vectors are linearly independent. {(1, 0, 2)T , (−1, 1, −1)T } is a basis for this space and its dimension is 2. (ii) The row space of A is spanned by the vectors (1, 0, 2, 4, −3) and (0, 1, 0, 4, 0). These vectors are linearly independent. {(1, 0, 2, 4, −3), (0, 1, 0, 4, 0)} is a basis for this space and its dimension is 2. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

8. Inner Product Spaces Let V be a real vector space. An inner product on V is a function that assigns to each ordered pair of vectors u, v in V real number (u, v) satisfying the following properties: (u, u) ≥ 0, (u, u) = 0 if and only if u = 0v (v, u) = (u, v) for any u, v in V (u + v, w) = (u, w) + (v, w) for any u, v, w in V (cu, v) = c(u, v) for u, v in V and c, a real scalar A real vector space that has an inner product defined on it is called an inner product space. If the space is finite dimensional it is called a Euclidean space. In an inner product space we define the length of a vector u by u = (u, u). Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

9. Gram-Schmidt Process Let V be an inner product space and W = {0} an m-dimensional subspace of V . Then there exists an orthonormal basis T = {w1, w2, · · · , wm} for W . Let S = {u1, u2, · · · , um} be any basis for W . Construct an orthogonal basis T∗ = {v1, v2, · · · , vm} for W. Select any one of the vectors in S, say u1 and call it v1. Look for a vector v2 in the subspace W1 of W spanned by {u1, u2} that is orthogonal to v1. v2 = u2 − (u2, v1) (v1, v1) v1 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

10. Gram-Schmidt Process Next, we look for a vector v3 in the subspace W2 of W spanned by {u1, u2, u3} that is orthogonal to both v1 and v2. v2 = u2 − (u2, v1) (v1, v1) v1 v3 = u3 − (u3, v1) (v1, v1) v1 − (u3, v2) (v2, v2) v2 Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

11. Example Let’s use the Gram-Schmidt process to ﬁnd an orthonormal basis for the subspace of R4 with basis u1 = (1, 1, 1, 0)T , u2 = (−1, 0, −1, 1)T and u3 = (−1, 0, 0, −1)T . First let v1 = u1, v2 = u2 − (u2, v1) (v1, v1) v1 =     −1 0 −1 1     − (− 2 3 )     1 1 1 1     =     −1/3 2/3 −1/3 1     Multiplying v2 by 3 to clear fractions, we get     −1 2 −1 3     Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

12. Example v3 = u3 − (u3, v1) (v1, v1) v1 − (u3, v2) (v2, v2) v2 =     −4/5 3/5 1/5 −3/5     Multiplying v3 by 5 to clear fractions, we get (−4, 3, 1, −3)T . An orthogonal basis: {v1, v2, v3} =     1 1 1 0     ,     −1 2 −1 3     ,     −4 3 1 −3     An orthonormal basis: {w1, w2, w3} =     1/ √ 3 1/ √ 3 1/ √ 3 0     ,     −1/ √ 15 2/ √ 15 −1/ √ 15 3/ √ 15     ,     −4/ √ 35 3/ √ 35 1/ √ 35 −3/ √ 35     Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

13. Factorization If A is an m × n matrix with linearly independent columns, then A can be factored as A = QR, Q: an m × n matrix whose columns form an orthonormal basis for the column space of A, R: an n × n nonsingular upper triangular matrix. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

14. Example Let’s ﬁnd the factorization of     1 −1 −1 1 0 0 1 −1 0 0 1 −1     . Let’s deﬁne the columns of A as the vectors u1, u2, u3. The orthonormal basis for the column space of A is w1 =     1/ √ 3 1/ √ 3 1/ √ 3 0     , w2 =     −1/ √ 15 2/ √ 15 −1/ √ 15 3/ √ 15     , w3 =     −4/ √ 35 3/ √ 35 1/ √ 35 −3/ √ 35     . Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

15. Example Q =     1/ √ 3 −1/ √ 15 −4/ √ 35 1/ √ 3 2/ √ 15 3/ √ 35 1/ √ 3 −1/ √ 15 1/ √ 35 0 3/ √ 15 −3/ √ 35     R =   r11 r12 r13 0 r22 r23 0 0 r33   where rji = (ui , wj ). R =   3/ √ 3 −2/ √ 3 −1/ √ 3 0 5/ √ 15 −2/ √ 15 0 0 7/ √ 35   A = QR Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

16. Linear Transformation A mapping L : V −→ W is said to be a linear transformation or a linear operator if L(αv1 + βv2) = αL(v1) + βL(v2) OR L(v1 + v2) = L(v1) + L(v2), (α = β = 1) L(αv) = αL(v) (v = v1, β = 0) Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

17. Example L(x) = 3x, x ∈ R2 . L(x + y) = 3(x + y) = L(x + y) L(αx) = 3(αx) = αL(x) L is a linear transformation. α : positive scalar F(x) = αx can be thought of as a stretching or shrinking by a factor of α. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

18. Example L(x) = x1e1, x ∈ R2 . If x = (x1, x2)T , then L(x) = (x, 0)T If y = (y1, y2)T , then αx + βy = αx1 + βy1 αx2 + βy2 L(αx + βy) = (αx1 + βy1)e1 = α(x1e1) + β(y1e1) = αL(x) + βL(y) L is a linear transformation, a projection onto the x1 axis. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

19. Example L(x) = (−x2, x1)T , x = (x1, x2)T ∈ R2. L(αx + βy) = −(αx2 + βy2) αx1 + βy1 = α −x2 x1 + β −y2 y1 = αL(x) + βL(y) L is a linear transformation. It has the eﬀect of rotating each vector in R2 by 90◦ in the counterclockwise direction. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

20. Example M(x) = (x2 1 + x2 2 )1/2, M : R2 −→ R. M(αx) = (α2 x2 1 + α2 x2 2 )1/2 =| α | M(x), αM(x) = M(αx), α < 0, x = 0. M is not a linear transformation. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

21. Linear One-to-one Transformations A linear transformation L : V → W is called one-to-one if it is a one-to-one function; that is, if v1 = v2 implies that L(v1) = L(v2). An equivalent statement is that L is one-to-one if L(v1) = L(v2) implies that v1 = v2. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

22. Linear Onto Transformations If L : V → W is a linear transformation of a vector space V into a vector space W , then the range of L or image of V under L, denoted by range L , consists of all those vectors in W that are images under L of vectors in V . Thus w is in range L if there exists some vector v in V such that L(v) = w. The linear transformation L is called onto if range L = W . Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

23. Linear Transformation and Singularity A is nonsingular. Ax = 0 has only the trivial solution. A is row (column) equivalent to In. For every vector b in Rn, the system Ax = b has a unique solution. det(A) = 0. The rank of A is n. The rows of A form a linearly independent set of vectors in Rn. The columns of A form a linearly independent set of vectors in Rn. The linear transformation L : Rn −→ Rn deﬁned by L(x) = A(x), for x in Rn, is one-to-one and onto. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

24. Similar Matrices If A and B are n × n matrices, we say that B is similar to A if there is a nonsingular matrix P such that B = P−1AP. Let V be any n−dimensional vector space and let A and B be any n × n matrices. Then A and B are similar if and only if A and B represent the same linear transformation L : V → V with respect to two ordered bases for V . If A and B are similar n × n matrices, then rank A = rank B. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

25. Example Let L : R3 → R3 be deﬁned by L([ u1 u2 u3 ]) = [ 2u1 − u3 u1 + u2 − u3 u3 ] and S = {[1 0 0], [0 1 0], [0 0 1]} be the natural basis for R3. The representation of L with respect of S is A =   2 0 −1 1 1 −1 0 0 1   . Considering S = {[1 0 1], [0 1 0], [1 1 0]} as ordered bases for R3, the transition matrix P from S to S is P =   1 0 1 0 1 1 1 0 0   P−1 =   0 0 1 −1 1 1 1 0 −1   . Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

26. Example Then the representation of L with respect to S is B = P−1 AP =   1 0 0 0 1 0 0 0 2   The matrices A =   2 0 −1 1 1 −1 0 0 1   and B =   1 0 0 0 1 0 0 0 2   are similar. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

27. References Linear Algebra With Applications, 7th Edition by Steven J. Leon. Elementary Linear Algebra with Applications, 9th Edition by Bernard Kolman and David Hill. Dr. Ceni Babaoglu cenibabaoglu.com Linear Algebra for Machine Learning: Factorization and Linear Transformations

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