- #1

gazzo

- 175

- 0

Let [itex]{\cal U}[/itex] be the subspace of [itex]\mathbb{R}_{2x2}[/itex] of all matrices of the form [tex]\left( \begin{array}{ccc}x&-x\\y&z\end{array}\right)[/tex].

Is it true, that

[tex]\left( \begin{array}{ccc}x&-x\\y&z\end{array}\right) = x\left( \begin{array}{ccc}1&-1\\0&0\end{array}\right) + y\left( \begin{array}{ccc}0&0\\1&0\end{array}\right) + z\left( \begin{array}{ccc}0&0\\0&1\end{array}\right) = x{\cal M}_1 + y{\cal M}_2 + z{\cal M}_3[/tex]

So [itex]{\cal B}=\left\{\cal M}_1,{\cal M}_2,{\cal M}_{3} \right\}[/itex] forms a basis for [itex]{\cal U}[/itex]. And that it has dimension 3 after showing that they are independent.

Also,

[tex]{\cal W} = \left( \begin{array}{ccc}a&b\\-a&c\end{array}\right)[/tex]

So

[tex]

\left\{\left( \begin{array}{ccc}1&0\\-1&0\end{array}\right),\left( \begin{array}{ccc}0&1\\0&0\end{array}\right),\left( \begin{array}{ccc}0&0\\0&1\end{array}\right)\right\}

[/tex]

is a basis for [itex]{\cal W}[/itex] (with three dimensions).

Any help appreciated. Thanks