4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below.-(:) -() --()v1 =V2 =V3 =and--(:) --E) --()1wi =, w2 =1, W3 =1Let L: R³ → R³ be the linear transformation defined byL:Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.

Answered: Image /qna-images/answer/d73f0db7-8fe9-4cd5-89d8-b69963e3377a.jpg

Answered: 4. Let B = {v1, v2, v3} and C = {w1,…

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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29. Let T be linear transformation from a vector space U (F) into a vector space r) and a, a2, ..., C, e then show that a,, a2, E U. If T(a), T (a), . . , T (a,) are linearly independent ..., An are linearly independent.
4. Let { u1, Uz, u3, U4, U5} be an orthogonal basis for R$ , y a vector in RS , w1 = Span { u1, Uz} and W, = Span{ u3, U4, Uz} . Which of the following is False? (a) W1 = W2" (b) W2 = W1' (c) There are two vectors zzin W1 and zzin W2such that y = z1 + z2 (d) y is orthogonal to W1 and W2.
7. Consider the set of vectors S = {u = (-1, 2, – 1), v = (2, – 5, 0), w = (1, – 3, – 1)}, then find the vector orthonormal projection of "u" on "v" and the vector orthonormal projection of "u" on "w" respectively.
8. Let A = | 1 -3 -5 -7 7 5 (a) Find the image of u = and define T: R³ R2 by T(x) = Ax. 2 1 under T. (b) Find a vector x whose image under T is b = -12 12 Explain why your work makes sense.
8. Show that {u,,,} is an orthogonal basis for R³. Then express x as a linear combination of the u's. 2₁ 2 -3 U₂ = 2 3 -=[3³] 5 ₁ and. =-3
3. (12) Let V = R². Define vector addition and scalar multiplication as follows: (u1, uz)O(v1, v2) = (u1V1, U2V2) and c(u1,u2) = (u,C, u2^), c > 0 (a) Does V have an additive identity? If so, what would it be? (b) Does the distributive axiom c(uOv) = cu@cv hold? (Show why or why not). 4
Suppose T: R³→R² is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(−U+2V). 2 U = 1 3 4 V = 3 4 0 T(-U+2V) = 0 0 -6 [:] 6 T(U)= -9 [3] 9 T(V) =
Suppose T: R¹ → R™ is a linear transformation and {V₁, V2, V3}) is a set of vectors in R". (a) Complete the following definition: The set {V1, V2, V3} is called linearly independent if (b) (c) T is one-to-one if and only if Ker (T) = Show that if T is one-to-one and {V₁, V2, V3} is linearly independent, then the set {T(V₁), T(V₂), T(V3)} is linearly independent.
Suppose T is a linear transformation, where T(u) T(7) (3, – 5) (1, 0, 0) (0, 1, 0) (0, 0, 1) (5, 4) 6- T(w) (5, 4) Find the following: T(5u + 70 + 2w) (Enter vectors (x, y) using ) T(( – 3, 5, 4)) = (Enter vectors (x, y) using ) -

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