3. LetW = (1,2,3,6), (4,-1,3,6), (5, 1, 6, 12))W2 = ((1,-1,1,1), (2, –1,4, 5))%3D%3Dbe subspaces of R4. Find bases for W1n W2 and W, + W2.Extend the basis of W1n W2 to a basis of W1 + W2.

Answered: Image /qna-images/answer/af612085-5789-4dba-a536-8e882f6b9f48.jpg

Answered: 3. Let W = ((1,2,3,6), (4,–1,3,6), (5,…

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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If V is the subspace spanned by (1, 1, 0, 1) and (0, 0, 1, 0), find (a) a basis for the orthogonal complement V-L. (b) the projection matrix P onto V. (c) the vector in V closest to the vector b = (0, 1, 0, -1) in V¹.
(a) Every vector of R² is a linear combination of the vectors (-2,3), (1, 1) and (4,1)? (b) Is every vector in R³a linear combination of the vectors (-2, 1, 3) and (3, 2, 1)? (c) The Vector (2, –1,2) belong to the subspace generated by vectors (3,2, -1) and (-3,0,2}?
6. Is the set S {(2,3, 1), (1,2, 1), (-2, 3, 3)} C R linearly %3D independent or dependent? Justify your answer. 7. Prove that the set S = {(1,3, 2), (3, 1, 2), (0, 1,0)} spans R. 8. Find a basis for R that includes the vectors (1,0, 2) and (0, 1, 1). Justify your answer. 9. Show that the function T: R → R defined by T(x, y, z) = (x – z, 2x + y, 2y – 3r) is a lincar transformation. 10. Define the function T: R3 → R as 2 1 T(v) = Av = -3 2 0 1 0- 0-1 Find the image of T(-4,5,1) and the preimage of (-1,2, 2). 11. Find the eigenvalues and the corresponding eigenvectors of the 2 matrix A 3
1. Determine whether the vectors (1,-1) and (1, 1)T are linearly inde- pendent or not. 2. Show that the vectors (1,0,0), (1, 1, 0) and (1,1,1) form a basis of R³. 3. Suppose that the vectors (1, 2, -1,0) and (1,3,2,0) span the sub- space U of R4. Determine whether the vector (1, 1, 1, 0) belongs to U or not.
Find a basis for the orthogonal complement of the subspace of Rª spanned by the following vectors. V1 = (1,-1,7,3), V2 = (2,-1,5,2), V3 = (1, 0, -2, -1) The required basis can be written in the form {(x, y, 1, 0), (z, w, 0, 1)}.
Q16 Please provide justified answer asap to get a upvote
Find the decomposition a = a|| + a₁ with respect to b if a = (3, 4, 2), b = (3, 4, 6). = (a||x, a||y, a||z) a¹b = (alx, aly, alz) (Enter the components of vectors using symbolic notation and fractions where needed.) a||x ally a||z ab = = = alx = aly = alz =
*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.
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
Let U and W be subspaces of R*; U=span{(1,1,0,-1), (1,2,2,-2), (1,2,3,0)} and W=span{(2,3,3,-1), (2,3,2,-3), (1,3,4,-3)}. Find a basis for U+W and dim(U+W).
3. Given A= [1 0 (a) Give the vector c so that Ax = b has a solution if and only if b - c = 0. (b) Give a basis of the subspace of all vectors b such that Ax = b has a solution. %3D

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3. Let W = ((1,2,3,6), (4,–1,3,6), (5, 1, 6, 12)) %3D W2 = ((1,-1,1,1), (2,-1,4, 5)) %3D be subspaces of R4. Find bases for W1n W2 and W, + W2. Extend the basis of W1n W2 to a basis of W1 + W2.