Homework Answers
υΣνΤ. Answer the following questions: Suppose a matrix A E Rmxn has an SVD A (i) Show that the rank of the miatrix A E Rmxn is equal to the number of its nonzero singular values. (ii) Show that miult...
υΣνΤ. Answer the following questions: Suppose a matrix A E Rmxn has an SVD A (i) Show that the rank of the miatrix A E Rmxn is equal to the number of its nonzero singular values. (ii) Show that miultiplication by an orthogonal matrix on the left and multiplication by an orthogonal matrix on the right, i.e., UA and BU, where A E Rmxn and B ERnm are general matrices, and U Rxm is an orthogonal matrix, preserve the Frobenius...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is inv...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
For the following problems use: Annx n matrix A is invertible RREF(A) = I rank(A) - n A 2 x 2 matrix A is invertible =...
For the following problems use: Annx n matrix A is invertible RREF(A) = I rank(A) - n A 2 x 2 matrix A is invertible = det(A) 0 3 singular (non-invertible). For which value(s) of h is A = -2 -1 -4 Choose... Choose... 6 2 h-2 a 0,b 0,c+0,d +0 A = 4 -1 C 0 x-2 or x 4 For which values of x is A = invertible a 0,b 0,c 0,d=0 4 x 2 X#1 and x2...
Let I, Y ER" be two nonzero n-dimensional vectors and define the n x n matrix...
Let I, Y ER" be two nonzero n-dimensional vectors and define the n x n matrix A = ty eigenvalues of A are 0 and y's Show that the
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such tha...
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
a e Octave a a Caleculator to qn i. Calculate the number N equal to the sum of all digits of your SID ii. For the N × N matrix A = (aij) defined as aij-l if i = j, or i + 1 = j, and aij 0 for all oth...
a e Octave a a Caleculator to qn i. Calculate the number N equal to the sum of all digits of your SID ii. For the N × N matrix A = (aij) defined as aij-l if i = j, or i + 1 = j, and aij 0 for all other values of (i,j) Compute (A-)x for x ,,1 ER . Compute (A-1)Tx for x = [1, 2, 3, . . . , NIT E RN. . Compute x-A(A-19%...
1 Let A = (4 22 a) Find elementary matrices E, Er Ez - such that...
1 Let A = (4 22 a) Find elementary matrices E, Er Ez - such that 2 E3 E₂E, A = I b) Find A
3. Let {an}nez+ be an Arithmetic Sequence with a nonzero common differ- ence. Let a =...
3. Let {an}nez+ be an Arithmetic Sequence with a nonzero common differ- ence. Let a = 1. Suppose that there exist a number 7 ER such that 2an+1 = yan +4, for all n e Zt. (a) (4 pts) Find y and the formula for an. azi_i, (notice that 21 - i is the sub- (b) (3 pts) Find the formula for script of a).
7.Let A be ann x n real symmetric invertible matrix, let B Rt and C E R. Define f:R R by 2 a. Giv...
I need help with a, b, and c.
7.Let A be ann x n real symmetric invertible matrix, let B Rt and C E R. Define f:R R by 2 a. Give f (a) c. Give f"(x) d. Prove that if A is positive definite and u is the critical point of f, then f(u) < f(x) for all x E Rn where x Prove that if A is negative definite and u is the critical point of f, then...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transfor...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
υΣνΤ. Answer the following questions: Suppose a matrix A E Rmxn has an SVD A (i) Show that the rank of the miatrix A E Rmxn is equal to the number of its nonzero singular values. (ii) Show that miultiplication by an orthogonal matrix on the left and multiplication by an orthogonal matrix on the right, i.e., UA and BU, where A E Rmxn and B ERnm are general matrices, and U Rxm is an orthogonal matrix, preserve the Frobenius...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
For the following problems use: Annx n matrix A is invertible RREF(A) = I rank(A) - n A 2 x 2 matrix A is invertible = det(A) 0 3 singular (non-invertible). For which value(s) of h is A = -2 -1 -4 Choose... Choose... 6 2 h-2 a 0,b 0,c+0,d +0 A = 4 -1 C 0 x-2 or x 4 For which values of x is A = invertible a 0,b 0,c 0,d=0 4 x 2 X#1 and x2...
Let I, Y ER" be two nonzero n-dimensional vectors and define the n x n matrix A = ty eigenvalues of A are 0 and y's Show that the
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
a e Octave a a Caleculator to qn i. Calculate the number N equal to the sum of all digits of your SID ii. For the N × N matrix A = (aij) defined as aij-l if i = j, or i + 1 = j, and aij 0 for all other values of (i,j) Compute (A-)x for x ,,1 ER . Compute (A-1)Tx for x = [1, 2, 3, . . . , NIT E RN. . Compute x-A(A-19%...
1 Let A = (4 22 a) Find elementary matrices E, Er Ez - such that 2 E3 E₂E, A = I b) Find A
3. Let {an}nez+ be an Arithmetic Sequence with a nonzero common differ- ence. Let a = 1. Suppose that there exist a number 7 ER such that 2an+1 = yan +4, for all n e Zt. (a) (4 pts) Find y and the formula for an. azi_i, (notice that 21 - i is the sub- (b) (3 pts) Find the formula for script of a).
I need help with a, b, and c.
7.Let A be ann x n real symmetric invertible matrix, let B Rt and C E R. Define f:R R by 2 a. Give f (a) c. Give f"(x) d. Prove that if A is positive definite and u is the critical point of f, then f(u) < f(x) for all x E Rn where x Prove that if A is negative definite and u is the critical point of f, then...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
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