Why is the basis of the orthogonal complement $U^\perp$ found by computing the null space of the row space of the subspace $U$?
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Because you are looking for all the vectors that are orthogonal to each of the vectors of the basis of your subspace $U$; hence when you multiply any vector in $U^\perp$ by any other vector in the basis of $U$, you should get $0$. This is equivalent to findind all $X$ such that $AX=0$, where $A$ is a matrix with the vectors of the basis of $U$ as rows. Hence why we solve fore the $\ker{A}$.
