Question

Let V be the vector space of all 2×2 matrices with real entries. Let H be the set of all 2×2 idempotent matrices with real entries. Is H a subspace of the vector space V? Does H contain the zero vector of V?

Answer 1

Remember, a set
`S\subset V`is a subspace of the vector space V if:

`1. 0\in S\\2.  a+b\in S,\text{ for all } a,b\in S\\3. \lambda a\in S, \text{for } \lambda \text{ a scalar}`.

1. Observe that

`0* 0=0^2=0`, where
`0` is the null matrix
`2* 2`.

Then
`0 \in H`

2. Consider two matrices A and B such that
`AB\\eq 0` in H

Observe that
`(A+B)^2=A^2+2AB+B^2\\=A+2AB+B\\eq A+B`.

Then,
`A+B\\subseteq H`

Then H is not a subspace of V