Question

Linear algebra: span?
Let W be the subspace of R4 defined by W = Span{(1, 0, 1, 0), (1, 1, 1, 1), (0, 1, 1, 1)}, and let S be the subset of R4 given by S = {(1, 0, 1, 0), (0, 1, 0, 1), (2, 3, 2, 3)}. Does span(S) = W? Justify your answer.

How am I supposed to know if span(S) = W? Do I row reduce W or S? All I know is that after row reduce the matrix, if the rows with all zeroes equals a certain letter(given that it's an augmented matrix with a, b, c, d), S would not span W.

I am really confused as to how to approach this question. Can someone please explain this to me?

Answer 1

Row reduce both sets of vectors

Answer 2

What do I do after row reducing both sets of vectors?

Answer 3

you should notice that they span different dimensions

Answer 4

span(w) is dimension 3 and span(S) is dimension 2

Answer 5

oh but we didn't learn dimensions yet. We're supposed to approach the question using the knowledge from span and subspaces.. is there another way?

Answer 6

show that (0,1,1,1) is not in the span of S

Answer 7

Oh I see. Thanks Zarkon. But how would I know when I get a question like this, whether it spans or not? would i pick a random vector from W and see if it is a linear combination of the vectors in S?

Answer 8

the Span of a set is a set. to show two sets are equal you need to show inclusion from both sides

ie show that

\[\text{span}(S)\subseteq \text{span}(W)\text{ and }\text{span}(W)\subseteq \text{span}(S)\]