Span

[hkus10]

Let S = {w1, w2, ..., wn} be a set of n vectors in R^n and let A be nxn matix whoise columns are the elements of S. Prove that for all b belong in R^n, Ax = b is consistent if and only if b belongs span(S).

 

My approach is:

I use contrapositive method to prove both sides

First, I prove that if Ax = b is consistent, the b belongs span(s).

 

Assume that Ax = b is inconsistent. Let x be [x1 x2 ... xn] ***This is a vectical vector which means x1, x2, and xn lines up vectically since I cannot express it in this way.

This means the last row of all vectors in S are zeros and the last row of b has nonzero integer. Then, b cannot be written as a linear combination of vectors in S since 0x1 + 0x2 + ... + 0xn = 0.

Therefore, b does not belong span(S).

 

For the other side:

Assume b does not belong span(S).

Then, b cannot be written as a linear combination of vectors in S.

If the last row of b is an nonzero integer, then the last row of all vectors in S must be zeros so that b cannot be written as a linear combination of vectors in S.

By the Matrix-Vector Product written in terms of columns, [v1 v2 ... vn][x] not equal to .

Thus, Ax = b is inconsistent.

 

My question is that this proof seems quite reasonably for me. However, am I really proving this question. If not, how to approach it instead?

 

Thanks