2. Let B = {v₁,..., Vn} be a subset of Fn containing n distinct vectors and let A = [V₁ V₂] bethe matrix having the vį as columns. Prove that B is a basis for F if and only if det (A) = 0.

Let B = {v1,..., vn} be a subset of Fn containing n distinct vectors and let A = [v1, v2,... vn] be…

Answered: 2. Let B = {V₁,..., Vn} be a subset of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find a basis for the set of all vectors of the form below, where a, b, c are real numbers. [ a-2b+5c, 2a+5b-8c, -a-4b+7c, 3a+b+c]
1. (a) Determine whether the following three vectors v, = (0, 3, 1,–1), V2 = (6, 0, 5, 1) and v3 = (4, -7, 1, 3) in R* are linearly dependent or linearly independent. (b) Is it possible to express v, as a linear combination of vz and v3? 2. Find the bases for the null space, row space and column space of the following matrix: 1 4 9. 3 A = -1 -2 4 -1 -1 -2 -1 2 7 8. Hence mention the rank and nullity of A. 3. Find the eigenvalues and the corresponding eigenvectors of the following matrix: 1 -3 3 A = 3 -5 3 \6 -6 4/ Find a matrix P that diagonalizes A. Hence calculate A7. 4. Find the steady-state vector q of the following regular transition matrix of a Markov chain by (a) iterative method (considering an initial vector) and (b) (0.90 0.15 0.10) linear system technique: P = 0.05 0.75 0.05 0.05 0.10 0.85/ 5. Determine whether the following three vectors v, = (1, 2, 3, 4), V2 = (0, 1, 0, –1) and vz = (1, 3, 3, 3) form a basis for R* or not ?
{a, ,a2.a3} be bases for a vector space V, and suppose a, = 4b1 - b2, a2 = - b, + 6b2 + b3, az = b2 - 3b3. Let A = and B = a. Find the change-of-coordinates matrix from A to B. b. Find [x]B for x = 4a1 + 5a2 + a3. ... 4 - 1 a. P = - 1 1 BEA 1 - 3 b. [x]B (Simplify your answer.)
Suppose matrix A has the SVD A = UEVT. The matrix U has exactly three columns, u₁, u2, u3. The first left singular vector of A is and the second singular vector of A is Determine the entries of the third left singular vector u3 such that Assume that $1, 82, and s3 are all positive numbers. --0 WN WON W 3 -0 U2 = 3 2 -2 ܒܐ ܟܘ ܂ ܩܘ ܟܬ ܚܘ --(3)
Let V be the set of all vectors of the form 4r – 6s – 6t r + 3s – 15t 11r + 7s + 8t 14r + 11s – 6t. with r, s and t real. Find a matrix A such that W = Col(A). A =
Compute the norm of the vector 7 = |||
Asap
Let A = {d1, d,}, B= {B, 6} be two bases of a vector space V. Suppose that 5 b1+7b; Let also 1 Then the change-of-coordinates matrix PA-B from the basis A to the basis B and [x]a are given by:
If u is a nonzero vector in R", then uu" is a rank One matrix. O True O False
Do both parts
16. Show that if the non-zero vectors s), s2),..., s) are conjugate to a positive definite matrix G, then the vectors are independent.

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2. Let B = {V₁,..., Vn} be a subset of Fn containing n distinct vectors and let A = [v₁... Vn] be the matrix having the v; as columns. Prove that B is a basis for F if and only if det (A) + 0.