91Ó°ÊÓ

Find the dual code \(C^{\perp}\) of the code \(C.\) \(C=\left\\{\left[\begin{array}{l}0 \\ 0 \\\ 0\end{array}\right],\left[\begin{array}{l}1 \\ 1 \\\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 0 \\\ 1\end{array}\right],\left[\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right]\right\\}\)

Orthogonally diagonalize the matrices by finding an orthogonal matrix Q and a diagonal matrix \(D\) such that \(Q^{T} A Q=D.\) \(A=\left[\begin{array}{llll}2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\\ 1 & 0 & 0 & 2\end{array}\right]\)

If \(b \neq 0\), orthogonally diagonalize \(A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]\).

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

Determine whether the given orthogonal set of vectors is orthonormal. If it is not, normalize the vectors to form an orthonormal set. $$\left[\begin{array}{l} \frac{3}{5} \\ \frac{4}{5} \end{array}\right],\left[\begin{array}{r} -\frac{4}{5} \\ \frac{3}{5} \end{array}\right]$$

Find the dual code \(C^{\perp}\) of the code \(C.\) \(C=\left\\{\left[\begin{array}{l}0 \\ 0 \\ 0 \\\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 1 \\ 0 \\\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 1 \\ 0 \\\ 1\end{array}\right],\left[\begin{array}{l}0 \\ 0 \\ 0 \\\ 1\end{array}\right]\right\\}\)

Find an orthogonal basis for \(R^{3}\) that contains the vector \(\left[\begin{array}{l}3 \\ 1 \\ 5\end{array}\right].\)

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

Find the dual code \(C^{\perp}\) of the code \(C.\) \(C=\left\\{\left[\begin{array}{l}0 \\ 0 \\ 0 \\ 0 \\\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 1 \\ 1 \\ 0 \\\ 1\end{array}\right],\left[\begin{array}{l}1 \\ 0 \\ 0 \\ 1 \\\ 0\end{array}\right],\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1 \\\ 1\end{array}\right]\right\\}\)

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