The orthogonal complement of a column space

Proposition (The orthogonal complement of a column space). Let A be a matrix and let \(W=\operatorname{Col}(A)\). Then

\[W^{\perp}=\operatorname{Nul}\left(A^T\right) . \]

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Proof:
To justify the first equality, we need to show that a vector \(x\) is perpendicular to the all of the vectors in \(W\) if and only if it is perpendicular only to \(v_1, v_2, \ldots, v_m\). Since the \(v_i\) are contained in \(W\), we really only have to show that if \(x \cdot v_1=x \cdot v_2=\cdots=x \cdot v_m=0\), then \(x\) is perpendicular to every vector \(v\) in \(W\). Indeed, any vector in \(W\) has the form \(v=c_1 v_1+c_2 v_2+\cdots+c_m v_m\) for suitable scalars \(c_1, c_2, \ldots, c_m\), so

\[\begin{aligned} x \cdot v &=x \cdot\left(c_1 v_1+c_2 v_2+\cdots+c_m v_m\right) \\ &=c_1\left(x \cdot v_1\right)+c_2\left(x \cdot v_2\right)+\cdots+c_m\left(x \cdot v_m\right) \\ &=c_1(0)+c_2(0)+\cdots+c_m(0)=0 . \end{aligned} \]

Therefore, \(x\) is in \(W^{\perp}\).
To prove the second equality, we let

\[A=\left(\begin{array}{c} -v_1^T- \\ -v_2^T- \\ \vdots \\ -v_m^T- \end{array}\right) \]

By the row-column rule for matrix multiplication in Section \(2.3\), for any vector \(x\) in \(\mathrm{R}^n\) we have

\[A x=\left(\begin{array}{c} v_1^T x \\ v_2^T x \\ \vdots \\ v_m^T x \end{array}\right)=\left(\begin{array}{c} v_1 \cdot x \\ v_2 \cdot x \\ \vdots \\ v_m \cdot x \end{array}\right) \text {. } \]

Therefore, \(x\) is in \(\operatorname{Nul}(A)\) if and only if \(x\) is perpendicular to each vector \(v_1, v_2, \ldots, v_m\)