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

[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.

 

[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

 

[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 $$<br /> W=Sp\{\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 1 \\ 2 & 2 & 3 \end{array} \right), \left( \begin{array}{ccc} -1 & 1 & -1 \\ -3 & -2 & -3 \end{array} \right) \}$$ are liner dependent, so I have to find 4 more independent matrices.

 

[lanedance]

ok, so what's the issue?

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