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Let W- ((x. y, 2)|x-2y 32) Vector addition and scalar multiplication are defined by (x, y.z) + (u, v, w)- (x+u. y+v.-w) k (x. y, ) = (kx, ky, k'z) Determine whether W is a subspace of R.if so, prove it. If not, explain why it is not a subspace.

Let W- ((x. y, 2)|x-2y 32) Vector addition and scalar multiplication are defined by (x, y.z) + (u, v, w)- (x+u. y+v.-w) k (x. y, ) = (kx, ky, k'z) Determine whether W is a subspace of R.if so, prove it. If not, explain why it is not a subspace.

Advanced Engineering Mathematics
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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Let W- ((x. y, 2))x- 2y = 32) Vector addition and scalar multiplication are defined by (x. y.z) + (u, v, w) - (x + u. y+w-w) k (x, y, a) - (kx, ky, k'2) Determine whether W is a subspace of R.if so, prove it. If not, explain why it is not a subspace. 2. (a) By inspection (withaut any calculation) determine whether the following set is independent (b) Determine whether the vectors u= (1,-3,7), v- (3,-1-1), and w (2,4,-5) in R form a basis set for R. Determine whether the polynomials u=2+2x+ 4zx,1+, and w 3+ 2x + 5x span P, (the set of polynomials of degree 2 or less). If not find the space spanned by the given vectors. 3. Let S- (v, vz, vy, v4), where 4. and v Find a basis for the space spanned by the space span(5) which is a subset of S. 5. -1 1 The eigenvalues of the matrix A1 2. 1 are -1-2, and 3. Also, the eigenvector corresponds to the eigenvalues are respectively and (a) Find the matrix P that diagonalizes A (b) Find the diagonal matrix D (c) Hence compute A0
Transcribed Image Text:Let W- ((x. y, 2))x- 2y = 32) Vector addition and scalar multiplication are defined by (x. y.z) + (u, v, w) - (x + u. y+w-w) k (x, y, a) - (kx, ky, k'2) Determine whether W is a subspace of R.if so, prove it. If not, explain why it is not a subspace. 2. (a) By inspection (withaut any calculation) determine whether the following set is independent (b) Determine whether the vectors u= (1,-3,7), v- (3,-1-1), and w (2,4,-5) in R form a basis set for R. Determine whether the polynomials u=2+2x+ 4zx,1+, and w 3+ 2x + 5x span P, (the set of polynomials of degree 2 or less). If not find the space spanned by the given vectors. 3. Let S- (v, vz, vy, v4), where 4. and v Find a basis for the space spanned by the space span(5) which is a subset of S. 5. -1 1 The eigenvalues of the matrix A1 2. 1 are -1-2, and 3. Also, the eigenvector corresponds to the eigenvalues are respectively and (a) Find the matrix P that diagonalizes A (b) Find the diagonal matrix D (c) Hence compute A0
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