# Direct Sum of two Subspaces

• Apr 15th 2010, 10:19 AM
charikaar
Direct Sum of two Subspaces
let U be subspace of \$\displaystyle R^4\$ spanned by the vectors
\$\displaystyle u_1=(1,-2,5,-3),u_2=(2,3,1,-4),u_3=(3,8,-3,-5)\$

the vectors \$\displaystyle v_1=(1,-2,5,-3),v_2(0,7,-9,2)\$ form basis for the Subspace U. and \$\displaystyle v_1,v_2,v_3=(0,0,1,0),v_4=(0,0,0,1)\$ form basis for \$\displaystyle R^4\$. How do i find the subspace W of \$\displaystyle R^4\$ such that \$\displaystyle R^4=U \oplus W\$

Thanks for any help.
• Apr 15th 2010, 12:01 PM
Drexel28
Quote:

Originally Posted by charikaar
let U be subspace of \$\displaystyle R^4\$ spanned by the vectors
\$\displaystyle u_1=(1,-2,5,-3),u_2=(2,3,1,-4),u_3=(3,8,-3,-5)\$

the vectors \$\displaystyle v_1=(1,-2,5,-3),v_2(0,7,-9,2)\$ form basis for the Subspace U. and \$\displaystyle v_1,v_2,v_3=(0,0,1,0),v_4=(0,0,0,1)\$ form basis for \$\displaystyle R^4\$. How do i find the subspace W of \$\displaystyle R^4\$ such that \$\displaystyle R^4=U \oplus W\$

Thanks for any help.

Hint: think orthogonal complement.
• Apr 16th 2010, 01:18 AM
charikaar
Still can't go ahead. anymore help/hints please?