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.
Definition: Given a subspace [latex]H[/latex] of [latex]\mathbb{R}^n[/latex], the orthogonal complement of [latex]H[/latex] is the set of vectors in [latex]\mathbb{R}^n[/latex], each of which is orthogonal to every vector in [latex]H[/latex]. We denote the orthogonal complement by [latex]H^{\perp}[/latex].
Definition: Consider an [latex]m\times n[/latex] matrix [latex]A[/latex] with entries [latex]a_{ij}[/latex]. The transpose of [latex]A[/latex], written [latex]A^{T}[/latex], is the [latex]n\times m[/latex] matrix with entries [latex][A^T]_{ji} =a_{ij}[/latex]. In other words, the rows of [latex]A[/latex] are columns of [latex]A^T[/latex] and the columns of [latex]A[/latex] are rows of [latex]A^T[/latex].
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].
