where \(\vec u_W\) is the closest vector to \(\vec u\) on \(W\) and \(\vec u_{W^\perp}\) is in \(W^\perp\)
Definition6.2.2.
Let \(W\) be a subspace of \(\IR^n\) and let \(\vec u\) be a vector in \(\IR^n\text{.}\) The orthogonal decompostion of \(\vec u\) is the decomposition of \(\vec u\) given by Fact 6.2.1. The orthogonal projection of \(\vec u\) is \(\vec u_W\text{.}\)
Activity6.2.3.
Let \(W\) be a the \(xy\)-plane in \(\IR^2\text{.}\)
(a)
have them find a simple orthogonal decomposition
Fact6.2.4.
Let \(T:\IR^n\rightarrow\IR^m\) be a linear transformation with standard matrix \(A\text{.}\) Let \(W\) be the image of \(T\text{,}\) that is, \(W\) is spanned by the columns of \(A\text{.}\) Then for any \(\vec u\) in \(\IR^m\text{,}\) the matrix equation
\begin{equation*}
A^TA\vec x=A^T\vec u
\end{equation*}
is consistent, and \(\vec u_W=A\vec x\) for any solution \(\vec x\text{.}\)
Observation6.2.5.
When the image of a linear transformation is one-dimensional, then its standard matrix \(A\) only has one column \(\vec v\text{,}\) and for any vector \(\vec u\text{,}\)
