Question

Linear algebra: span

S = {v1, v2, ..., vn} is a basis for a vector space V and T = {w1, w1, ..., wr} is a linearly independent set of vectors in V. (n and r are subscripts)

Let T1={w1, v1, ..., vn}. Since S spans V, so does T1.

I don't understand how T1 would span V just because S spans V? A vector from T, w1, was added in T1 but tt doesn't state that T spans V.. so how does T1 span V?

Answer 1

T1 contains all the basis vectors from S therefore it clearly will span V

Answer 2

So even though w1, an vector from T, is in T1?

Answer 3

"T = {w1, w1, ..., wr} is a linearly independent set of vectors in V"

the vectors in T are taken from V

Answer 4

Oh I see. Thanks Zarkon!

Answer 5

btw, I was wondering how w1 is a linear combination of the vectors in S?

Answer 6

since S is a basis for V and w1 is in V then w1 can be writen as a LC of the vectors in S