91Ó°ÊÓ

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

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

Use the Gram-Schmidt Process to find an orthogonal basis for the column spaces of the matrices. $$\left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$

Show that the given vectors form an orthogonal basis for \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\). Then use Theorem 5.2 to express. w as a linear combination of these basis vectors. Give the coordinate vector \([\mathbf{w}]_{8}\) of w with respect to the basis \(\mathcal{B}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) of \(\mathbb{R}^{2}\) or \(\boldsymbol{B}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\) of \(\mathbb{R}^{3}\). $$\mathbf{v}_{1}=\left[\begin{array}{l}1 \\\1 \\\1\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}2 \\\\-1 \\\\-1 \end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}0 \\\1 \\\\-1\end{array}\right] ; \mathbf{w}=\left[\begin{array}{l}1 \\\2 \\\3 \end{array}\right]$$

Orthogonally diagonalize the matrices by finding an orthogonal matrix \(Q\) and a diagonal matrix \(D\) such that \(Q^{T} A Q=D\) $$A=\left[\begin{array}{llll} 3 & 0 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 1 & 0 & 0 & 3 \end{array}\right]$$

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

Use the Gram-Schmidt Process to find an orthogonal basis for the column spaces of the matrices. $$\left[\begin{array}{rrr} 1 & 2 & -1 \\ 1 & 0 & -1 \\ 1 & 1 & 1 \\ 1 & 1 & 5 \end{array}\right]$$

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

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