91Ó°ÊÓ

Fill in the missing entries of \(Q\) to make \(Q\) an orthogonal matrix. $$Q=\left[\begin{array}{cccc} 1 / 2 & 2 / \sqrt{14} & * & * \\ 1 / 2 & 1 / \sqrt{14} & * & * \\ 1 / 2 & 0 & * & * \\ 1 / 2 & -3 / \sqrt{14} & * & * \end{array}\right]$$

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 \(C\). \(G=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \\ 0 & 1\end{array}\right]\)

In Exercises \(15-18,\) find the orthogonal projection of v onto the subspace \(W\) spanned by the vectors \(\mathbf{u}_{i}\). ( You may assume that the vectors \(\mathbf{u}_{i}\) are orthogonal. $$\mathbf{v}=\left[\begin{array}{r} 7 \\ -4 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

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

If \(A\) and \(B\) are orthogonally diagonalizable and \(A B=\) \(B A,\) show that \(A B\) is orthogonally diagonalizable.

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 \(C\). \(P=\left[\begin{array}{llll}1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1\end{array}\right]\)

Determine whether the given matrix is orthogonal. If it is, find its inverse. $$\left[\begin{array}{ll}0 & 1 \\\1 & 0\end{array}\right]$$

In Exercises \(15-18,\) find the orthogonal projection of v onto the subspace \(W\) spanned by the vectors \(\mathbf{u}_{i}\). ( You may assume that the vectors \(\mathbf{u}_{i}\) are orthogonal. $$\mathbf{v}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right]$$

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 \(C\). \(P=\left[\begin{array}{lllll}1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1\end{array}\right]\)

In Exercises \(15-18,\) find the orthogonal projection of v onto the subspace \(W\) spanned by the vectors \(\mathbf{u}_{i}\). ( You may assume that the vectors \(\mathbf{u}_{i}\) are orthogonal. $$\mathbf{v}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{r} 2 \\ -2 \\ 1 \end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r} -1 \\ 1 \\ 4 \end{array}\right]$$