91Ó°ÊÓ

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

Orthogonally diagonalize the matrices by finding an orthogonal matrix Q and a diagonal matrix \(D\) such that \(Q^{T} A Q=D.\) \(A=\left[\begin{array}{ll}4 & 1 \\ 1 & 4\end{array}\right]\)

Determine which sets of vectors are orthogonal. $$\left[\begin{array}{r} -3 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{l} 2 \\ 4 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ 2 \end{array}\right]$$

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. $$\mathbf{x}_{1}=\left[\begin{array}{l}1 \\\1\end{array}\right], \mathbf{x}_{2}=\left[\begin{array}{l}1 \\\2\end{array}\right]$$

Determine which sets of vectors are orthogonal. $$\left[\begin{array}{r} 4 \\ 2 \\ -5 \end{array}\right],\left[\begin{array}{r} -1 \\ 2 \\ 0 \end{array}\right],\left[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right]$$

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

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. $$\mathbf{x}_{1}=\left[\begin{array}{r} 3 \\ -3 \end{array}\right], \mathbf{x}_{2}=\left[\begin{array}{l} 3 \\ 1 \end{array}\right]$$

In Exercises \(1-6,\) find the orthogonal complement \(W^{\perp}\) of \(W\) and give a basis for \(W^{\perp}\) $$W=\left\\{\left[\begin{array}{l} x \\ y \end{array}\right]: 3 x+4 y=0\right\\}$$

Determine which sets of vectors are orthogonal. $$\left[\begin{array}{r} 3 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ 2 \\ 1 \end{array}\right],\left[\begin{array}{r} 2 \\ -2 \\ 4 \end{array}\right]$$

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