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# Can anyone help me with vector spaces? watch

1. Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in M(a 2x4 matrix) with the property that FA=0 (the zero matrix in N(a 3x4 matrix)). Determine if H is a subspace of M(the 2x4 matrix).

2. Well, what are the axioms H has to satisfy to be a subspace?
3. (Original post by DFranklin)
Well, what are the axioms H has to satisfy to be a subspace?
1) It needs to include zero
2) the addition needs to be closed
3) the scalar multiplication needs to be closed

But I don't know how to answer the question by using these.
4. Well, is 0 in H or not?
5. (Original post by DFranklin)
Well, is 0 in H or not?
How can I know that?
6. Let A =0. Does A have "the property that FA=0 (the zero matrix in N(a 3x4 matrix))"?
7. (Original post by DFranklin)
Let A =0. Does A have "the property that FA=0 (the zero matrix in N(a 3x4 matrix))"?
Yes, because if FA=0 then A can be equal to zero... A = 0F^(-1) = 0
8. So, 0 is in H, so that's the first axiom checked.

For the 2nd axiom: if , is ? That is, if FA = 0 and FB=0, does F(A+B) = 0?

Similarly for the 3rd axiom.
9. (Original post by DFranklin)
So, 0 is in H, so that's the first axiom checked.

For the 2nd axiom: if , is ? That is, if FA = 0 and FB=0, does F(A+B) = 0?

Similarly for the 3rd axiom.
The second axiom is true, because if FA=0 and FB=0 then F is probably a zero matrix...so F(A+B)=0 right?

For the thrid axiom, cFA = 0 is the same as FA=0 because we can divide by c on both sides of the equation...is that right?

Thank you very much.
10. (Original post by Artus)
The second axiom is true, because if FA=0 and FB=0 then F is probably a zero matrix...so F(A+B)=0 right?
No. You should realize that anything like "so F is probably..." is never going to be a valid proof - but in this case it's completely wrong. F doesn't have to be zero.

For the thrid axiom, cFA = 0 is the same as FA=0 because we can divide by c on both sides of the equation...is that right?
Not quite. You need to be considering F(cA), not c(FA). Again, you need to use one of the standard rules for matrix multiplication.

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