6.2 Orthogonal complements and the matrix transpose (17/12) -- Math 220, Matrices

6.2 Orthogonal complements and the matrix transpose Reading Try out the Preview Activity and read Orthogonal complements and the matrix transpose in Understanding Linear Algebra by David Austin. If the matrix A has columns [latex]\vec{v}_1, \dots, \vec{v}_n[/latex], then [latex]A^T\vec{x} = [\begin{pmatrix} \vec{v}_1 \cdot \vec {x} \\ \vdots \\ \vec{v}_n \cdot \vec {x} \end{pmatrix}[/latex]. We see that we can compute all n dot products by finding this product. [latex]Nul(A^T) = (Col(A))^{\perp}[/latex]. Proof: [latex]A^T \vec{x} = \vec{0}[/latex] if and only if [latex]\vec{x}[/latex] is orthogonal to every column of [latex]A[/latex]. The columns of [latex]A[/latex] span [latex]Col(A)[/latex], so [latex]\vec{x}[/latex] is orthogonal to all element of [latex]Col(A)[/latex].