Linear Algebra Answers

Consider the vector space V = C

2 with scalar multiplication over the real numbers R and let W

and U be the subspaces of V defined by

W = {(z1, z2) ∈ V : z2 = z1 + 2z1} and U = {(z1, z2) ∈ V : z2 = z1 − z1}.

2.1 Find a basis for W ∩ U.

2.2 Express (z1, z2) ∈ V as (z1, z2) = w + u where w ∈ W and u ∈ U.

2.3 Explain whether V = W ⊕ U

Consider the vector space V = C

2 with scalar multiplication over the real numbers R and let

W = {(z1, z2) ∈ V : z2 = z1 + 2z1}.

1.1 Use the Subspace Test to show that W is a subspace of V.

1.2 Find a basis for W.

1.3 Explain whether W is a subspace over C

Linear transformation T : P2 → P2, T(a+bx+cx2 ) = (2a−b)+(a+b−3c)x+(c−a)x 2

a. Find the matrix representation A of T with respect to the ordered bases B = {1, x + 1, x2 + 1} and B0 = {1, x, x2}.

b. Let q(x) = a + bx + cx2 ∈ P2. Verify that A[q(x)]B = [T(q(x))]B'.

 Let S be a subset of F3 defined as S = {(x, y, z) € F3 = x +y + 2x – 1=0}. Then determine S is a subspace of F3 or not.

Let S be a subset of F3 defined as S =  (x; y; z) F 3 : x+y+2z-1=0

is S a subspace of F3 or not