91Ó°ÊÓ

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

Orthogonally diagonalize the matrices by finding an orthogonal matrix Q and a diagonal matrix \(D\) such that \(Q^{T} A Q=D.\) \(A=\left[\begin{array}{rr}9 & -2 \\ -2 & 6\end{array}\right]\)

In Exercises \(1-6,\) find the orthogonal complement \(W^{\perp}\) of \(W\) and give a basis for \(W^{\perp}\) $$W=\left\\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right]: 2 x-y+3 z=0\right\\}$$

Determine which sets of vectors are orthogonal. $$\left[\begin{array}{l} 5 \\ 3 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ -2 \\ 1 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ -1 \end{array}\right]$$

The given vectors form a basis for \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\). Apply the Gram-Schmidt Process to obtain an orthogonal basis. Then normalize this basis to obtain an orthonormal basis. $$\mathbf{x}_{1}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right], \mathbf{x}_{2}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \mathbf{x}_{3}=\left[\begin{array}{l} 1 \\ 0 \\ 0\end{array}\right]$$

Orthogonally diagonalize the matrices by finding an orthogonal matrix Q and a diagonal matrix \(D\) such that \(Q^{T} A Q=D.\) \(A=\left[\begin{array}{lll}5 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 3 & 1\end{array}\right]\)

The given vectors form a basis for \(a\) subspace \(W\) of \(\mathbb{R}^{3}\) or \(\mathbb{R}^{4}\). Apply the Gram-Schmidt Process to obtain an orthogonal basis for \(W\). $$\mathbf{x}_{1}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right], \mathbf{x}_{2}=\left[\begin{array}{l} 3 \\ 4 \\ 2 \end{array}\right]$$

The given vectors form a basis for \(a\) subspace \(W\) of \(\mathbb{R}^{3}\) or \(\mathbb{R}^{4}\). Apply the Gram-Schmidt Process to obtain an orthogonal basis for \(W\). $$\mathbf{x}_{1}=\left[\begin{array}{r} 2 \\ -1 \\ 1 \\ 2 \end{array}\right], \mathbf{x}_{2}=\left[\begin{array}{r} 3 \\ -1 \\ 0 \\ 4 \end{array}\right]$$

\(P\) is a parity check matrix for a code \(C\) Bring P into standard form and determine whether the corresponding code is equal to \(C.\) \(P=\left[\begin{array}{llll}1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1\end{array}\right]\)

Determine which sets of vectors are orthogonal. $$\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 1 \\ 2 \end{array}\right]$$