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Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the orthogonal complement for W in $\Bbb{R}^4$.

(a) Find orthonormal bases for $W$ and $W^\bot$.

I found $W=span \{(3,-2,-0,1),(0,1,1,0)\} \;$ by evaluating $$ \left[ \begin{array}{cccc|c} 1&1&-1&-1&0\\ 0 & 1 & -1 & 2&0 \end{array} \right] $$

but what I don't understand is why span of $W^\bot$ are the two rows of the row-echelon matrix i.e. $W^\bot=span \{(1,1,-1,-1),(0,1,-1,2)\} \;$. I've spent hours but can't figure out the connection of $W^\bot$and those two rows. Please help.

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The row space of a matrix is the orthogonal complement of its null-space. $\boldsymbol{Ax}$ can be written as a column of inner products of each row with $x$. Therefore, $\boldsymbol{Ax} = 0$ iff $\boldsymbol{x}$ is orthogonal to each row of the matrix.

You reduce it to row-echelon form just to ensure you don't include dependent rows in your basis.

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