Special subspace of M(2*3) (R)

1. Aug 23, 2010

estro

$$W=Sp\{\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 1 \\ 2 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} -1 & 1 & -1 \\ -3 & -2 & -3 \end{array} \right) \}$$

I have to find subspace T, so that $$M_{2*3}(R)=W\oplus T$$

I solved it by finding 5 liner independent matrices (in relation to matrices in W) and made them basis for T.

I'll appreciate any ideas.

2. Aug 23, 2010

lanedance

not sure if I'm reading it correctly but don't you need to find the matricies orthogonal to those in W

you will need 6 linearly independent matricies to span M_2,3

3. Aug 23, 2010

estro

Maybe I should clear myself.

I have to find a liner space T so that, $$T+W=M_{2*3}$$ and $$T \cap W=0$$

I've noticed that matricies from $$W=Sp\{\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 1 \\ 2 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} -1 & 1 & -1 \\ -3 & -2 & -3 \end{array} \right) \}$$ are liner dependent, so I have to find 4 more independent matrices.

4. Aug 23, 2010

lanedance

ok, so whats the issue?

note if it helps you can wirte them as 6-vectors and do normal gram-schimdt...