Let V = R4 and let U be defined below:U =x, y E Rx + y(a) Prove that U is a subspace of V.(b) Calculate the dimension of U by finding a basisfor U.(c) Give an example of a subspace W of V such thatW has dimension 1 andW OU = {0}.

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Answered: Let V = Rª and let U be defined below:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let u = 1 0 -6 9 a basis for W¹. Answer: V = 1 -1 3 J and let W the subspace of R4 spanned by {u, v}. Find ہے
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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
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Let V = Rª and let U be defined below: U = x, y E R x + y (a) Prove that U is a subspace of V. (b) Calculate the dimension of U by finding a basis for U. (c) Give an example of a subspace W of V such that W has dimension 1 and W OU = {0}.