When \(S\) is a line in \(\R^2\) then the projection of \(\v\) to \(S\) is the shadow of \(\v.\)
1.
Sketch the projection of \(\v=\begin{pmatrix} 4\\3 \end{pmatrix}\) onto \(S=\text{span} \left\{ \begin{pmatrix} 6\\0 \end{pmatrix} \right\}\) above. What is the value of \(\P\v?\)
2.
What is the error vector \(e_{S}?\)
3.
Verify that \(e_S \perp \begin{pmatrix} 6\\0 \end{pmatrix}.\)
Exercise Group.
Let \(\vec{b}=\left(\begin{array}{c}5\\2 \end{array} \right)\) and \(S=\)span\(\left\{\left(\begin{array}{r}-3\\7 \end{array} \right)\right\}\text{.}\)
4.
Project \(\b\) to \(S\) by solving the normal equation as explained in Remark 7.1.39 (1.) to find \(\boldsymbol{\mathcal{P}}\vec{b}.\)
5.
Find the associated error vector \(\vec{e}\) and check that \(\vec{e} \perp \left(\begin{array}{r}-3\\7\end{array}\right).\)
Verify the values found in Problem 7.2.1.4: that is that \(P\b=\boldsymbol{\mathcal{P}}\b\) and \(\vec{e}=(I-P)\b.\)
Exercise Group.
9.
Solve the normal equations to find the projection of \(\left(\begin{array}{r} 3\\-1 \end{array} \right)\) to span\(\left\{\left(\begin{array}{r}2\\3 \end{array} \right)\right\}.\)
10.
Find the projection matrix \(P\) for Problem 7.2.1.9 and verify that \(P^2=P.\)
Exercise Group.
11.
Solve the normal equations to find the projection of \(\left(\begin{array}{r}-3\\2\\0 \end{array} \right)\) to the column space of \(A=\left(\begin{array}{rr}1\amp -1\\-2\amp 2\\1\amp 1 \end{array} \right)\text{.}\)
12.
Find the projection matrix \(P\) for Problem 7.2.1.11 and verify that \(P^2=P\text{.}\)
13.
Project \(\w=\left(\begin{array}{r}4\\-1\\3 \end{array} \right)\) to span\(\left\{\left(\begin{array}{r}-1\\1\\2 \end{array} \right),\left(\begin{array}{r}2\\3\\1 \end{array} \right)\right\}\) by solving the normal equation as explained in Remark 7.1.39 (1.) to find \(\boldsymbol{\mathcal{P}}\w\text{.}\)
Verify the value found in Problem 7.2.1.13; that is \(P\w=\boldsymbol{\mathcal{P}}\w.\)
17.
Use the method given in Remark 7.1.39 (2.) to show that the projection matrix for projecting the vector \(\v=\begin{pmatrix} a\\b\\c \end{pmatrix}\) to the \(xy\)-plane in \(\R^3\) is as given in Example 7.1.29.
18.
Find the least-squares line for the points \((-2,3),\,(1,6)\) and \((3,6)\) and sketch the points and the line on the same graph.
