Consider the set of matrices W a a b a b b a b R Show tha

Consider the set of matrices W = {[a a + b a - b b]: a, b R}. Show that W is a subspace of M_22.

Solution

The set of S, 2x2 symmetric matrices must satisfy the following conditions in order to be a subspace the vector space M of all 2x2 matrices
A) W M
B) W is a vector space

In practical terms, this can be shown by demonstrating that W satisfies the following four conditions. Keep in mind that W is the set of matrices of the form
a...a+b
a-b....c

1) W M : Trivially true
2) 0 W
0...0
0...0 is symmetric, so 2 holds
3) If u W and v W, then u + v S
Let u =
a...a+b
a-b....c
and v =
d...e
e...f
u + v =
a+d...a+b+e
a-b+e......b+f is symmetric, so 3 holds
4) If u W and g is a scalar, then gu W
gu=
ga...g(a+b)
g(a-b)....gb which is symmetric, so 4 holds
Therefore S is a subspace of M

u =
a1...0 +
..0...0
b0...1+
..1...0
c0...0
..0...1.
So these three matrices form a basis for W

Similar reasoning holds for the skew symmetric case

so W is a vectorspace of M22