# rowspace and kernel

by UrbanXrisis
Tags: kernel, rowspace
 P: 1,214 $$A = \left(\begin{array}{cccc}-1 &6&5&9 \\ -1&0&1&3 \end{array}\right)$$ Find orthonormal bases of the kernel, row space. To find the bases, I did reduced the array to its RREF. $$A = \left(\begin{array}{cccc}1 & 0&-1&-3\\ 0&1&2/3&1 \end{array}\right)$$ Then the orthonormal bases would just be that divided by the length. $$||v_1||=\sqrt{1+1+3^2}=\sqrt{11}$$ $$||v_2||=\sqrt{1+(2/3)^2+1}=\sqrt{2.44444}$$ so that means, the orthonormal bases would be: $$A = \left(\begin{array}{cccc} \frac{1}{ \sqrt{11}} & 0&\frac{-1}{ \sqrt{11}}&\frac{-3}{ \sqrt{11}} \\0 & \frac{1}{ \sqrt{2.44444}} & \frac{.66666}{ \sqrt{2.44444}} &\frac{1}{ \sqrt{2.44444}}\end{array}\right)$$ what exactly is the orthonormal bases of the kernel? Also, isnt the row space the same as the vectors of the bases? I think I also did something wrong in my calculations
 Sci Advisor P: 1,253 If you are looking for an orthonormal basis, one thing to check is that your basis actually is orthogonal. The basis you have found for the row space is not orthogonal. Given an arbitrary basis for a vector space, do you know how to construct an orthogonal basis for it via the Gram-Schmidt process?
 P: 1,214 is what i did above the orthonormal row space? that is wrong as well, i dont know why.... however: using the Gram-Schmidt process, i still get an error: $$A = \left(\begin{array}{cccc}-1 &6&5&9 \\ -1&0&1&3 \end{array}\right)=\left(\begin{array}{cc}W_1 &W_2 \end{array}\right)$$ want to find an orthonormal basis $$R={U_1 ,U_2}$$ $$U_1=\frac{W}{||W_1||}$$ $$||W_1||=\sqrt{11}$$ $$U_1=\frac {\left(\begin{array}{cccc}-1 &6&5&9 \end{array}\right)}{\sqrt{11}}$$ $$U_2=\frac{W_2- U_1}{||W_2- U_1||}$$ where $$W_2=\left(\begin{array}{cccc} -1&0&1&3 \end{array}\right)$$ $$U_1=\frac {\left(\begin{array}{cccc}-1 &6&5&9 \end{array}\right)}{\sqrt{11}}$$ $$U_1=\left(\begin{array}{cccc}-1/\sqrt{11} &6\sqrt{11}&5\sqrt{11}&9\sqrt{11} \end{array}\right)$$ is the the correct set up to get an orthonormal basis? Also, to get an orthonormal row space, what would I have to do?