Problem 14.Determine if the statements are true or false.??1. If the set of vectors U spans a subspace S, then vectors can be added to U to create a basis for S2. The zero vector is a basis for the zero subspace3. The matrix representation of a linear transformation from R¹ to R³ is 3 x 44. If S=span{u), u₂, us}, then dim(S) = 3

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Answered: Problem 14. Determine if the statements…

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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4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
Question 10 Find the specified change-of-coordinates matrix. Consider two bases B = {b₁, b₂} and C = {₁, c2} for a vector space V such that b₁ = c₁ - 5c2 and by = 2c₁-4c2. Fi coordinates matrix from B to C. O 0 27 -5-4 [12] 024 Question 11
0 -8 3 Let B= {b,, b2, b3} be a basis for a vector space V. Find T(3b, - 5b,) when T is a linear transformation from V to V whose matrix relative to B is [T)R =0 2 -7 7 -6 2 ..... T(3b, - 5b2) =
13. The set of solutions to the system of linear equations x12x2 + x3 0 2x13x2 + x3 0 is a subspace of R³. Find a basis for this subspace. =
5. Find the standard matrix of the given linear transformation: (a) P: R? - R? projects a vector onto the line y = -2x. For the vector u = find P(ü). (b) Rg: R3 - R rotates a vector clockwise about the z - axis through the angle 30°. For the 21 vector u = 2| find R9 (ũ). 6. For the following linear transformations find S • T and T • S. Indicate the domain and codomain of the new linear transformations. x+ y+z] x-y-zl 2x + 7. Find the inverse of the following linear transformation: [3x + 2y] [5x + 3yl %3D
4. Find the vector x determined by the given coordinate vector [x]g and the given basis В. 8. [x]s = B = -5
198. Apply the 57-degree counterclockwise rotation about the origin to the vectors [1,0] and [0, 1], then use the image vectors (written as columns) to form the coefficient matrix M for the rotation. Test M by calculating the products M and M Where does this rotation send the vector [3, 1]? Does M, when applied to 3 do its job correctly? 6. 6.
*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.
9. Consider the following 2 x 2 matrix A and basis S of R²: A ¹ = [² %) and 5 = {[_¹₂] · [_³;]} The matrix A defines a linear operator on R². Find a matrix B that represents the mapping A relative to the basis S.
3.4. Find the distance from a vector (2,3, 1)" to the subspace spanned by the vectors (1, 2,3)T, (1,3,1)T. Note, that I am only asking to find the distance to the subspace, not the orthogonal projection.
Dd.58.
21. Suppose that A {a1,a2} is a basis of the vector space V = R? and B, = a1 + 2a2, B2 = 201 + 5a2. (a) Prove that B = {B,,B,} is also a basis of V. %3D (b) Find the following coordinate matrices : [B1]A, [B,]B, [a1]a and [a2]B. (c) Find the transition matrix (=Change of basis matrix) from A to B

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