Let W be the subspace of R4 spanned by S V₂ = (1, 1, 1, −1), Let V₁ =(2, 1, 2, 1), W₁ = (1, 1, 0, 3), 1, V2, V3, V4}, where v3 = (5,3,5,1), V₁ = (-1, -2, -1, 4). W2 = = (0, 1, 0, -3). (a) Find a square matrix A whose column space is W. (b) Determine whether w₁ and we are in W. Justify your answers. (c) Find a subset B of S which is a basis of W.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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a b c 

Let W be the subspace of Rt spanned by S
1, V2, V3, V4}, where
V1 =(2, 1, 2, 1),
V2 = (1,1, 1, –1), V3
(5,3, 5, 1),
V4 = (-1, –2, –1, 4).
Let
Wi =(1, 1,0, 3),
W2 = (0, 1, 0, –3).
(a) Find a square matrix A whose column space is W.
(b) Determine whether wị and w2 are in W. Justify your answers.
(c) Find a subset B of S which is a basis of W.
(d) Express each of the vectors in S\ B as a linear combination of the vectors in B.
(e) What is the dimension of W?
(f) Find a basis for W+, the orthogonal complement of W.
Transcribed Image Text:Let W be the subspace of Rt spanned by S 1, V2, V3, V4}, where V1 =(2, 1, 2, 1), V2 = (1,1, 1, –1), V3 (5,3, 5, 1), V4 = (-1, –2, –1, 4). Let Wi =(1, 1,0, 3), W2 = (0, 1, 0, –3). (a) Find a square matrix A whose column space is W. (b) Determine whether wị and w2 are in W. Justify your answers. (c) Find a subset B of S which is a basis of W. (d) Express each of the vectors in S\ B as a linear combination of the vectors in B. (e) What is the dimension of W? (f) Find a basis for W+, the orthogonal complement of W.
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