91Ó°ÊÓ

(a) If \(\|A\|\) is an operator norm, prove that \(\|I\|=1\) where I is an identity matrix. (b) Is there a vector norm that induces the Frobenius norm as an operator norm? Why or why not?

\(\langle\mathbf{u}, \mathbf{v}\rangle\) is an inner product. In Exer. cises \(31-34\), prove that the given statement is an identity. $$\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}=\frac{1}{2}\|\mathbf{u}+\mathbf{v}\|^{2}+\frac{1}{2}\|\mathbf{u}-\mathbf{v}\|^{2}$$

Let \(\|A\|\) be a matrix norm that is compatible with a vector norm \(\|\mathbf{x}\|\). Prove that \(\|A\| \geq|\lambda|\) for every eigenvalue \(\lambda\) of \(A\).

\(\langle\mathbf{u}, \mathbf{v}\rangle\) is an inner product. In Exer. cises \(31-34\), prove that the given statement is an identity. $$\langle\mathbf{u}, \mathbf{v}\rangle=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}$$

Find \(\operatorname{cond}_{1}(A)\) and \(\operatorname{cond}_{\infty}(A)\). State whether the given matrix is ill-conditioned. $$A=\left[\begin{array}{ll}3 & 1 \\\4 & 2\end{array}\right]$$

Find the plane \(z=a+b x+c y\) that best fits the data points \((0,-4,0),(5,0,0),(4,-1,1),(1,-3,1),\) and (-1,-5,-2)

Find \(\operatorname{cond}_{1}(A)\) and \(\operatorname{cond}_{\infty}(A)\). State whether the given matrix is ill-conditioned. $$A=\left[\begin{array}{rr}1 & -2 \\\\-3 & 6\end{array}\right]$$

Find the standard matrix of the orthogonal projection onto the subspace \(W\). Then use this matrix to find the orthogonal projection of \(\mathbf{v}\) onto \(W\). $$W=\operatorname{span}\left(\left[\begin{array}{l} 1 \\ 1 \end{array}\right]\right), \mathbf{v}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right]$$

Find \(\operatorname{cond}_{1}(A)\) and \(\operatorname{cond}_{\infty}(A)\). State whether the given matrix is ill-conditioned. $$A=\left[\begin{array}{ll}1 & 0.99 \\\1 & 1\end{array}\right]$$

Find the standard matrix of the orthogonal projection onto the subspace \(W\). Then use this matrix to find the orthogonal projection of \(\mathbf{v}\) onto \(W\). $$W=\operatorname{span}\left(\left[\begin{array}{r} 1 \\ -2 \end{array}\right]\right), \mathbf{v}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$