linear algebra
Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a...
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Let u, v, w be three vectors in R4 with the property that 4u - 30+2w = 0. Let A be the 4 x 2 matrix whose columns are u and u (in that order). Find a solution to the equation Ac =W. Let 1 -2 0 3 A=1 -2 2-1 2 -4 1 4 Find a list of vectors whose span is the set of solutions to Ax = 0. 1 1 Enter the list...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> W be a linear transformation. (a) Prove that for every pair of ordered bases B = exists a unique m x n matrix A such that [T(E)]c = A[r3 for all e V. The matrix A is called the (B,C)-matrix of T, written A = c[T]b. (b) For each n E N, let Pm be the vector space of...
ANSWER SHOULD BE NEAT CLEAN AND WELL EXPLAINED.HANDWRITTEN NEAT
CLEAN,EACH STEP SHOULD BE EXPLAINED WELL
Find the M to meet the Lyapunov equation in (3.59) with What are the eigenvalues of the Lyapunov equation? Is the Lyapunov equation singular? Is the solution unique? Repeat Problem 3.31 for B- Ci- A1 -2 with two different C 3.7 Lyapunov Equation Consider the equation AM +MB C (3.59) where A and B are, respectively, n x n andmx m constant matrices. In order...
Question 6 (2 pts). [Exercise 4.1.9] Let V = W = R 2 . Choose
the basis B = {x1, x2} of V , where x1 = (2, 3), x2 = (4, −5) and
choose the basis D = {y1, y2} of W, where y1 = (1, 1), y2 = (−3,
4). Find the matrix of the identity linear mapping I : V → W with
respect to these bases.
QUESTION 6 (2 pts). Exercise 4.1.9 Let V = W...
5. Given a linear map f R3R3 if V Vi, V2, va) is a basis of R3, and further, a) State the defining matrix of f under the basis vi, V2, vs) -3 2 0 b) Let W-(w1, w2, w3) be another basis of R3 and P42 be the change- 01-1 of-coordinate matrix from V to W. Let A be the defining matrix for f under the basis W diagonalize A.
5. Given a linear map f R3R3 if V...
Let {v1, v2, ..., vn} and {w1, w2, ..., wn} be bases of V and W , respectively. Prove: ∃ ! α ∈ Hom(V, W ) s.t. ∀ i ∈ {1, 2, ..., n}, α(vi) = wi
please show steps
Let 0 a12 a13 a14 0 a34 a42 023 a43 0 a14 a31 a24 a41 0 a12 a32 a13 a21 0 a21 0 a2 a2 a31 a32 0 a34 be two antisymmetric matrices, where ak -aki, or ATA and BT -B. Show that AB BA and present this diagonal matrix as follows BA AB (a32014 +a13024 a21a34) I, where I is the 4 x 4-identity matrix. Find A-1 and B-1. (H. Minkowski, 1908)
Let 0 a12 a13...
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L :V + W be a linear transformation, and let Ker L = {2 € V: L(x)=0}. (a) If Ker L = {0}, show that C = {L(v1), L(02), ..., L(vn) } CW is a linearly independent set in W. (b) If C = {L(01), L(V2),..., L(Un)} C W is a linearly independent set in W,...
How can I get the (a) 3*2 matrix A?
x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...