Hi everyone,

I've just got a question today. I've given it a few attempts, but I keep getting stuck, and I'm not sure where I'm going wrong.

Let S and T be subspaces of $\displaystyle {{M}_{2}}(\mathbb{R})$, where:

S is generated by the matrices $\displaystyle \{ {A_1},{A_2},{A_3}\} $, where $\displaystyle {A_1} = \left( {\begin{array}{*{20}{c}} 0&1\\ 0&1 \end{array}} \right),{A_2} = \left( {\begin{array}{*{20}{c}} 0&1\\ 1&1 \end{array}} \right),{A_3} = \left( {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right)$

and T is generated by the matrices $\displaystyle \{ {B_1},{B_2}\} $, where $\displaystyle {B_1} = \left( {\begin{array}{*{20}{c}} 1&1\\ 1&0 \end{array}} \right),{B_2} = \left( {\begin{array}{*{20}{c}} 2&1\\ 1&1 \end{array}} \right)$.

I've already done the first part of the question, determining dim(S) and dim(T) [Just had to prove the matrices contained are linearly independent.]

The part I need help with is determining the basis and dimension of $\displaystyle S \cap T$.

I've tried creating a spanning set, and then proving that they're linearly independent, and it indicates that they are. Which doesn't make sense, because then it implies that the dimension theorem is wrong.

I figure I'll throw it out there, I can't find any examples on the internet for anything similar, and I'm only getting more confused.