Videos for Section 6.6


So far we have learned how to project a vector onto a subspace if we have an orthogonal basis for the subspace. But what if we have an arbitrary basis? We could convert the arbitrary basis to an orthogonal basis using Gram-Schmidt, but there's an easier way. The lone video for this section explains how.

If $W$ is a subspace of $V$ spanned by vectors $\{{\bf b_1}, \ldots {\bf b_k}\}$, and if we want to decompose a vector ${\bf x}$ as $${\bf x} = c_1 {\bf b}_1 + \cdots + c_k {\bf b_k} + {\bf x}_\perp,$$ where ${\bf x}_\perp$ is orthogonal to all of the ${\bf b}_j$'s, then we take inner products to set up a system of equations $$\begin{pmatrix} \langle {\bf b}_1 | {\bf b}_1\rangle & \cdots & \langle {\bf b_1} | {\bf b_k} \rangle \cr \vdots & \ddots & \vdots \cr \langle {\bf b_k} | {\bf b_1} \rangle & \cdots & \langle {\bf b_k} | {\bf b_k} \rangle \end{pmatrix} \begin{pmatrix} c_1 \cr \vdots \cr c_k \end{pmatrix} = \begin{pmatrix} \langle {\bf b_1} | {\bf x} \rangle \cr \vdots \cr \langle {\bf b_k} | {\bf x} \rangle \end{pmatrix},$$ which we can solve by row-reduction. This is particularly simple if $V=R^n$ with the standard inner product. In that case, we package the vectors ${\bf b}_j$ into a matrix $$A = \begin{pmatrix} \phantom{x} && \cr {\bf b_1} & \cdots & {\bf b_k} \cr \phantom{x}&& \end{pmatrix}$$ and our equations reduce to $$A^T A {\bf c} = A^T {\bf x}$$