Question

Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 matrices with real entries that have trace 0. Is H a subspace of the vector space V?

Answer 1

tr(zA+B) = za+zb + a1+b1 = z(a+b)+(a1+b1) = 0+0 = 0 .

Explanation:

You need to show that given two matrices A,B such that tr(A) = tr(B) = 0 and a random number "z", tr(zA+B) = 0.

`A = \left[\begin{array}{cc}a&x\\y&b\end{array}\right] \\B = \left[\begin{array}{cc}a1&x1\\y1&b1\end{array}\right] \\`

By hypothesis a+b=0 and a1+b1 = 0

tr(zA+B) = za+zb + a1+b1 = z(a+b)+(a1+b1) = 0+0 = 0

Therefore H would be a subspace of V.

Answer 2

H is not a subspace of v

Explanation:

Please see attachment