Linear Algebra ( Subspaces )

Question

Let vector w = (1, 1, 2, -1).

W = { u ∈ R^4 | u ⋅ w = 2 }, and V = { u - (1, 0, 1, 1) | for all u ∈ W }.

(a) Is W a subspace? Explain your answer.

(b) Show that V is a subspace and that T = { (-1,1,0,0), (-2,0,1,0), (1,0,0,1) } is a basis for V.

(c) Let T be the basis of V defined in (b). Find [(4,-1,1,5)]_T. Show your workers, however, you do not need to show your elementary row operations.

Thank you!

Answer 1

(a). W is not a subspace of R⁴ since the zero vector (0,0,0,0) is not in W.

(c). Find real numbers a, b, c so that a(-1,1,0,0) + b(-2,0,1,0) + c(1,0,0,1) = (4,-1,1,5)

Equating components, we have:

-a - 2b + c = 4

a + 0 + 0 = -1

0 + b + 0 = 1

0 + 0 + c = 5

So, a = -1, b = 1 and c = 5.