91Ó°ÊÓ

Find an orthogonal basis for \(\mathbb{R}^{4}\) that contains the vectors $$\left[\begin{array}{r}2 \\\1 \\\0 \\\\-1\end{array}\right] \text { and }\left[\begin{array}{l}1 \\\0 \\\3 \\\2\end{array}\right]$$

Determine whether the given orthogonal set of vectors is orthonormal. If it is not, normalize the vectors to form an orthonormal set. $$\left[\begin{array}{l} \frac{1}{2} \\ \frac{1}{2} \end{array}\right],\left[\begin{array}{r} \frac{1}{2} \\ -\frac{1}{2} \end{array}\right]$$

Fill in the missing entries of \(Q\) to make Q an orthogonal matrix. $$Q=\left[\begin{array}{ccc} 1 / \sqrt{2} & 1 / \sqrt{3} & * \\ 0 & 1 / \sqrt{3} & * \\ -1 / \sqrt{2} & 1 / \sqrt{3} & * \end{array}\right]$$

In Exercises \(11-14\), let \(W\) be the subspace spanned by the given vectors. Find a basis for \(W^{\perp}\) $$\mathbf{w}_{1}=\left[\begin{array}{r} 2 \\ -1 \\ 6 \\ 3 \end{array}\right], \mathbf{w}_{2}=\left[\begin{array}{r} -1 \\ 2 \\ -3 \\ -2 \end{array}\right], \mathbf{w}_{3}=\left[\begin{array}{l} 2 \\ 5 \\ 6 \\ 1 \end{array}\right]$$

Either a generator matrix \(G\) or a parity check matrix \(P\) is given for a code \(C .\) Find a generator matrix \(G^{\perp}\) and a parity check matrix \(P^{\perp}\) for the dual code of \(C.\) \(G=\left[\begin{array}{ll}1 & 1 \\ 1 & 1 \\ 1 & 0 \\ 0 & 1\end{array}\right]\)

Determine whether the given orthogonal set of vectors is orthonormal. If it is not, normalize the vectors to form an orthonormal set. $$\left[\begin{array}{l} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{array}\right],\left[\begin{array}{r} \frac{2}{3} \\ -\frac{1}{3} \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ -\frac{5}{2} \end{array}\right]$$

Determine whether the given orthogonal set of vectors is orthonormal. If it is not, normalize the vectors to form an orthonormal set. $$\left[\begin{array}{r} \frac{1}{2} \\ \frac{1}{2} \\ -\frac{1}{2} \\ \frac{1}{2} \end{array}\right],\left[\begin{array}{c} 0 \\ \frac{1}{3} \\ \frac{2}{3} \\ \frac{1}{3} \end{array}\right],\left[\begin{array}{r} \frac{1}{2} \\ -\frac{1}{6} \\ \frac{1}{6} \\ -\frac{1}{6} \end{array}\right]$$

If \(A\) is an invertible matrix that is orthogonally diagonalizable, show that \(A^{-1}\) is orthogonally diagonalizable.

In Exercises \(11-14\), let \(W\) be the subspace spanned by the given vectors. Find a basis for \(W^{\perp}\) $$\mathbf{w}_{1}=\left[\begin{array}{r} 4 \\ 6 \\ -1 \\ 1 \\ -1 \end{array}\right], \mathbf{w}_{2}=\left[\begin{array}{r} 1 \\ 2 \\ 0 \\ 1 \\ -3 \end{array}\right], \mathbf{w}_{3}=\left[\begin{array}{r} 2 \\ 2 \\ 2 \\ -1 \\ 2 \end{array}\right]$$

Either a generator matrix \(G\) or a parity check matrix \(P\) is given for a code \(C .\) Find a generator matrix \(G^{\perp}\) and a parity check matrix \(P^{\perp}\) for the dual code of \(C.\) \(G=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \\ 0 & 1\end{array}\right]\)