i have a question about subspaces, is the following subset (lets call it W)
a subspace of (let's call it V)
$\displaystyle W=\{A\in V |A^T = -A\}$
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 $\displaystyle \alpha, \forall A, B \in W, \alpha A + B \in W$
Idea: Observe that $\displaystyle (\alpha A + B)^T = \alpha A^T + B^T =-(\alpha A + B) $
So.....?
Do you know that W is a subspace of V iff for any scalar $\displaystyle \alpha, \forall A, B \in W, \alpha A + B \in W$?
Its easy to prove, just check all the axioms by an appropriate choice of alpha.
The idea shows that $\displaystyle \alpha A + B$ is of the form that belongs to W.
$\displaystyle W=\{A\in V |A^T = -A\}
$
means any element, say A, in W has to satisfy $\displaystyle A^T = -A$. It does not mean that it is the only thing in the set. So if B is in W, then$\displaystyle B^T = -B$.
To prove that a subset W is a subspace you need to show:
$\displaystyle A\in W$ and $\displaystyle B\in W \implies A+B \in W$
For any scalar $\displaystyle \alpha, A\in W \implies \alpha A \in W$
W is non-empty.
For the first statement, assume that $\displaystyle A^T = -A$ and $\displaystyle B^T = -B$ and show that $\displaystyle (A+B)^T = -(A+B)$. 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 $\displaystyle A^T = -A$.