91Ó°ÊÓ

Find the projection of \(b\) onto the column space of \(A\) : $$ A=\left[\begin{array}{rr} 1 & 1 \\ 1 & -1 \\ -2 & 4 \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 2 \\ 7 \end{array}\right] $$ Split \(b\) into \(p+q\), with \(p\) in the column space and \(q\) perpendicular to that space. Which of the four subspaces contains \(q\) ?

Find a vector \(x\) orthogonal to the row space of \(A\), and a vector \(y\) orthogonat to the column space, and a vector \(z\) orthogonal to the nullspace: $$ A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 2 & 4 & 3 \\ 3 & 6 & 4 \end{array}\right] $$

Find a basis for the orthogonal complement of the row space of \(A\) : $$ A=\left[\begin{array}{lll} 1 & 0 & 2 \\ 1 & 1 & 4 \end{array}\right] $$ Split \(x=(3,3,3)\) into a row space component \(x_{r}\) and a nullspace component \(x_{n}\).

If \(P\) is the projection matrix onto a line in the \(x-y\) plane, draw a figure to describe the effect of the "reflection matrix \(^{\prime \prime} H=I-2 P\). Explain both geometrically and algebraically why \(H^{2}=I\).

If \(P_{C}=A\left(A^{\top} A\right)^{-1} A^{\top}\) is the projection onto the column space of \(A\), what is the projection \(P_{R}\) onto the row space? (It is not \(\left.P_{C}^{\mathrm{T}} !\right)\).

What is the closest function \(a \cos x+b \sin x\) to the function \(f(x)=\sin 2 x\) on the interval from \(-\pi\) to \(\pi\) ? What is the closest straight line \(c+d x\) ?

Construct a matrix with the required property or say why that is impossible. (a) Column space contains \(\left[\begin{array}{r}1 \\ 2 \\\ -3\end{array}\right]\) and \(\left[\begin{array}{r}2 \\ -3 \\\ 5\end{array}\right]\), nullspace contains \(\left[\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right]\). (b) Row space contains \(\left[\begin{array}{r}1 \\ 2 \\\ -3\end{array}\right]\) and \(\left[\begin{array}{r}2 \\ -3 \\\ 5\end{array}\right]\), nullspace contains \(\left[\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right]\). (c) \(A x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]\) has a solution and \(A^{T}\left[\begin{array}{l}1 \\ 0 \\\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]\). (d) Every row is orthogonal to every column ( \(A\) is not the zero matrix). (e) The columns add up to a column of 0 s, the rows add to a row of \(1 \mathrm{~s}\).