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Section 6.2 Projections (ON2)
Learning Outcomes
Subsection 6.2.1 Class Activities
Fact 6.2.1 .
Let \(W\) be a subspace of \(\IR^n\) and let \(\vec u\) be a vector in \(\IR^n\text{.}\) Then we can write \(\vec u\) uniquely as
\begin{equation*}
\vec u=\vec u_W+\vec u_{W^\perp}
\end{equation*}
where \(\vec u_W\) is the closest vector to \(\vec u\) on \(W\) and \(\vec u_{W^\perp}\) is in \(W^\perp\)
Definition 6.2.2 .
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
\(\vec u_W\text{.}\)
Activity 6.2.3 .
Let \(W\) be a the \(xy\) -plane in \(\IR^2\text{.}\)
(a) have them find a simple orthogonal decomposition
Fact 6.2.4 .
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 \(\vec x\text{.}\)
Exercises 6.2.3 Exercises
