1.
Solve the normal equations to find the projection of \(\left(\begin{array}{r} 5\\-2 \end{array} \right)\) to span\(\left\{\left(\begin{array}{r}1\\4 \end{array} \right)\right\}\text{.}\)
Section 7.3 Fundamentals of Projections: Homework
Exercises Exercises
2.
Solve the normal equations to find the projection of \(\left(\begin{array}{r}-1\\2\\5\end{array}\right)\) to span\(\left\{\left(\begin{array}{r}2\\-2\\1\end{array} \right),\left(\begin{array}{r}-1\\-1\\1 \end{array}\right)\right\}.\)
3.
Find the projection of \(\left(\begin{array}{r}1\\4\\27\end{array}\right)\) to span\(\left\{\left(\begin{array}{r}3\\4\\1\end{array} \right),\left(\begin{array}{r}-2\\-2\\6 \end{array}\right)\right\}.\)
4.
Find the projection of \(\left(\begin{array}{r}4\\2\\-12\end{array}\right)\) to span\(\left\{\left(\begin{array}{r}1\\0\\3\end{array} \right),\left(\begin{array}{r}-5\\-2\\9 \end{array}\right)\right\}.\)
5.
Find the projection of \(\left(\begin{array}{r}1\\-9\\4\\3\end{array}\right)\) to span\(\left\{\left(\begin{array}{r}2\\1\\1\\1\end{array} \right),\left(\begin{array}{r}-3\\2\\3\\3 \end{array}\right)\right\}.\)
6.
Find the projection of \(\left(\begin{array}{r}2\\-3\\2\\1\end{array}\right)\) to span\(\left\{\left(\begin{array}{r}1\\0\\-2\\2\end{array} \right),\left(\begin{array}{r}-5\\-2\\3\\-2 \end{array}\right)\right\}.\)
7.
Let \(P\) be a projection matrix (this is characterized by \(P^2=P\)). Prove that \(I-P\) is a projection matrix.
8.
Let \(P\) project to col\((A)\) and \(P'\) project to col\((2A)\text{.}\) What can you say, with justification from algebra and/or theorems, definitions, etc. - about the relationship between \(P\) and \(P'\text{?}\)
9.
Prove Theorem 7.1.4.
10.
Find the equation of the line closest to the set \(\{(1,1),(3,5),(5,8)\}\text{.}\) Plot the points and the line on the same axes (carefully - axes labeled and straight, line straight, etc.).
11.
Find the equation of the line closest to the set \(\{(1,1),(2,3),(3,7)\}\text{.}\) Plot the points and the line on the same axes (carefully - axes labeled and straight, line straight, etc.).
12.
Pick any specific point \((x_4,y_4)\) on the line you found in Problem 7.3.11. Include it as a fourth point in the set of points given in Problem 7.3.11, and then find the line closest to your set of four points. Explain your result.
13.
Find the equation of the line closest to \(\{(1,1),(2,4),(5,13)\}\text{.}\) Plot the points and the line on the same axes (carefully - axes labeled and straight, line straight, etc.).
14.
Let \(n\in\N\) satisfy \(n\ge2\) and consider the set \(\{(0,0),(1,0),(n,n)\}\) of points in \(\R^2\text{.}\) Find the equation of the line closest to this set, as a function of \(n\) of course. As \(n\ra+\infty\text{,}\) what happens to this line?
15.
Let \(n\in\N\) satisfy \(n\ge2\) and consider the set \(\{(0,0),(1,1),(n,0)\}\) of points in \(\R^2\text{.}\) Find the equation \(y=m(n)x+b(n)\) of the line closest to this set, as a function of \(n\) of course. As \(n\ra+\infty\text{,}\) what line does the line descirbed by \(y=m(n)x+b(n)\) converge to?
16.
Find the parabola (described by \(f(x)=ax^2+bx+c\)) closest to \(\{(-2,2),(-1,0),(1,1),(2,5)\}\text{.}\) Plot the points and the line on the same axes (carefully - axes labeled and straight, parabola neat, etc.).
17.
Find the parabola (described by \(f(x)=ax^2+bx+c\)) closest to \(\{(-1,2),(0,0),(1,1),(2,5)\}\text{.}\) Plot the points and the line on the same axes (carefully - axes labeled and straight, parabola neat, etc.).
Exercise Group.
Recall Definition 2.10.21. We can invert some nonsquare matrices from one side provided either \(\A^T\A\) or \(\A\A^T\) is invertible. Let \(\A=\in\R^{m\times n}.\)
18.
Prove that if \(\A^T\A\) is invertible then
\begin{equation*}
\A^+_L:=\left(\A\A^T\right)^{-1}\A^T
\end{equation*}
satisfies \(\A^+_L\A=\I_n.\) In this case \(\A^+_L\) is the left-pseudo-inverse of \(\A.\)19.
Prove that if \(\A\A^T\) is invertible then
\begin{equation*}
\A^+_R:=\A^T\left(\A\A^T\right)^{-1}
\end{equation*}
satisfies \(\A\A^+_R=\I_m.\) In this case \(\A^+_R\) is the right-pseudo-inverse of \(\A.\)20.
Prove that if \(\A\) is nonsingular then \(\A^+_L=\A\A^+_R=\A^{-1}.\)Exercise Group.
Refer to Exercise 7.3.18 or Exercise 7.3.19. Find whichever pseudo-inverse is defined each of the following matrices.
