# Math Help - Subset, Subspaces and Linear Algebra

1. ## Subset, Subspaces and Linear Algebra

i have a question about subspaces, is the following subset (lets call it W)
a subspace of (let's call it V)

$W=\{A\in V |A^T = -A\}$

and V is a two by two matix

hope someone understands

2. Originally Posted by iLikeMaths
i have a question about subspaces, is the following subset (lets call it W)
a subspace of (let's call it V)

$W=\{A\in V |A^T = -A\}$

and V is a two by two matix

hope someone understands

Question: If V is the vector space consisting of all two by two matrices, then is W a subspace??

Useful Fact: W is a subspace of V iff for any scalar $\alpha, \forall A, B \in W, \alpha A + B \in W$

Idea: Observe that $(\alpha A + B)^T = \alpha A^T + B^T =-(\alpha A + B)$

So.....?

3. could you explain a bit more please, i am totally clueless

4. Originally Posted by iLikeMaths
could you explain a bit more please, i am totally clueless
Do you know that W is a subspace of V iff for any scalar $\alpha, \forall A, B \in W, \alpha A + B \in W$?

Its easy to prove, just check all the axioms by an appropriate choice of alpha.

The idea shows that $\alpha A + B$ is of the form that belongs to W.

5. where did $B$ come from i thought we were dealing with $A^T and -A$

6. Originally Posted by iLikeMaths
i have a question about subspaces, is the following subset (lets call it W)
a subspace of (let's call it V)

$W=\{A\in V |A^T = -A\}$

and V is a two by two matix

hope someone understands
Originally Posted by Isomorphism
Do you know that W is a subspace of V iff for any scalar $\alpha, \forall A, B \in W, \alpha A + B \in W$?

Its easy to prove, just check all the axioms by an appropriate choice of alpha.

The idea shows that $\alpha A + B$ is of the form that belongs to W.
Originally Posted by iLikeMaths
where did $B$ come from i thought we were dealing with $A^T and -A$
$W=\{A\in V |A^T = -A\}
$

means any element, say A, in W has to satisfy $A^T = -A$. It does not mean that it is the only thing in the set. So if B is in W, then $B^T = -B$.

7. can any one provided a step by step explaination to the original question because i really need to understand this? do i find the matrix A, please help

8. To prove that a subset W is a subspace you need to show:
$A\in W$ and $B\in W \implies A+B \in W$
For any scalar $\alpha, A\in W \implies \alpha A \in W$
W is non-empty.

For the first statement, assume that $A^T = -A$ and $B^T = -B$ and show that $(A+B)^T = -(A+B)$. Use a similar technique for the second statement. You could also do these two in a single step if you follow Isomorphism's method. For the last statement, you simply need to find a 2 by 2 matrix that satisfies $A^T = -A$.

9. the zero vector satisfies $A^T = -A$ hence W is a subspace of V, is this correct

10. the zero vector satisfies hence W is a subspace of V, is this correct
It is certainly a start. But to conclude that W is a subspace of V you need to prove all 3 of the statements I listed before or prove Isomorphism's statement and do the bit you have already done.