Basis for the vector space of idempotents

[yifli]

Homework Statement

A matrix a is idempotent if a^2=a. Find a basis for the vector space of all 2x2 matrices consisting entirely of idempotents

2. The attempt at a solution
the vector space in question is dimension 4, so I need to find 4 idempotent matrices.
but i don't want to find them by 'guessing'. Is there a good way to find them?

Thanks

 

[Mark44]

Homework Statement

A matrix a is idempotent if a^2=a. Find a basis for the vector space of all 2x2 matrices consisting entirely of idempotents

2. The attempt at a solution
the vector space in question is dimension 4, so I need to find 4 idempotent matrices.
but i don't want to find them by 'guessing'. Is there a good way to find them?

I'm guessing that the dimension of the subspace of idempotent 2 x 2 matrices is less than 4.

I would work with the equation A² = A, and see what I could come up with about a matrix in this space.

It might also be helpful to look at an arbitrary matrix, such as
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Note that in every text I've ever seen, matrices are represented by capital letters; e.g., A, B, C, and so on.