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 /> <br /> W=Sp\{\left( \begin{array}{ccc} 1 &amp; 1 &amp; 1 \\ 1 &amp; 2 &amp; 3 \end{array} \right), \left( \begin{array}{ccc} 1 &amp; 0 &amp; 1 \\ 2 &amp; 2 &amp; 3 \end{array} \right), \left( \begin{array}{ccc} -1 &amp; 1 &amp; -1 \\ -3 &amp; -2 &amp; -3 \end{array} \right) \}<br /> 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...