9.W.1 The Gram matrix of an inner product on R2 with respect to the standard basis is G=[1 223Find thegram matrix of the same inner product with respect to the basis(2][3)

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Answered: 9.W.1 The Gram matrix of an inner…

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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Suppose A is the matrix for T: R3 R³ relative to the standard basis. Find the diagonal matrix A' for T relative to 1 A'= 2 0 0 1 -1 0 0 0 -1 B'= {(1, 1, 0), (2, 1, 0), (0, 0, 1)} A = -2 0 0 0 1 0 11 F 0 0 1
C - 3 [2 [00] Consider the ordered bases B = (621) and (2) [ -5 = 5 3 for the vector space V of lower triangular 2 × 2 matrices with zero trace. a. Find the transition matrix from C to B. ТВ b. Find the coordinates of M in the ordered basis B if the coordinate vector of M in C is [M]c [M]B= c. Find M. M =
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A square matrix U E CNN is unitary if UTU = UU = I, where I is the NX N identity matrix. Prove that a square matrix is unitary if and only if the rows and columns of U form an orthonormal basis for CN.
(2, 4, 0), the Gram-Schmidt process Given vị = (-1, 2, –1), v2 = (1, 7,1) and v3 allows us to construct an orthonormal basis {u1, u2, U3 } of R³ such that u¡ is a multiple of V1, u2 is a linear combination of vj and v2, and uz is a linear combination of v1, V2 and v3. Then u2 is equal to: O (1,1, 1) O (1,2, 0) (1:2'0뚜 이 O (0, 1, 0) o (-1, 2, –1) ㅇ 늘(1,0, 1)
*1. Let W be the subspace of R5 spanned by the vectors B Find a basis for W. Give the dimension of W. -3 b₁ = 2 *2. Consider B = {b₁,b₂}, a basis for R² where - (²). b₂ = (3) 0 (a) Find the change of basis matrix that converts vectors given with respect to B into vectors given with respect to the standard basis. Give your matrix explicitly. (b) Find the change of basis matrix that converts vectors given with respect to the standard basis into vectors given with respect to B. Give your matrix explicitly. 3. True or False? (a) There is a 5 x 3 matrix whose column space is 1-dimensional. (b) If A is an m x n matrix, then rank(A) + nullity(A) = m. (c) If A is an n x n matrix and the columns of A span R", then A is invertible. (d) If B is a 2 x 2 matrix and B² = I₂, then B = I2. (e) The kernel of the function T: R² R² given by T + (²) - (² + 1) ₁ = is {0}. (f) There is a matrix that has a 3-dimensional column space and a 2-dimensional row space.
State if the following statement is true or false: If y is a vector space and BC y spansv, then B contains a basis for v. True False
Suppose A is the matrix for T: R³ → R³ relative to the standard basis. Find the diagonal matrix A' for T relative to the basis B'. 0-2 0 10 0 1 0), (2, 1, 0), (0, 0, 1)} A' = A = -1 0 B' = {(1, 1, 00.7 007 ↓↑
of Let W be the set of all 2 × 2 matrices [a b] C d basis for W is S = (a) • X (b) following questions: • Y = • r {[ || (c) where, c = 0 and d = 2a - 3b. A || •S= 1 x ] []} " S О У I dim(W) - Answer the
1 3. (а) Find an orthonormal basis B {ū1, ū2, đ3} of R³ such that u and the x-component of ūz is 1/3. [1 Find the coordinate vector of 3 relative to the basis B. (b) 1

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9.W.1 The Gram matrix of an inner product on R² with respect to the standard basis is G = -1 12 Find the gram matrix of the same inner product with respect to the basis { [2] 23 3