91Ó°ÊÓ

Find a least squares solution of \(A \mathbf{x}=\mathbf{b}\) by constructing and solving the normal equations. $$A=\left[\begin{array}{ll} 3 & 1 \\ 1 & 1 \\ 1 & 2 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]$$

Find a least squares solution of \(A \mathbf{x}=\mathbf{b}\) by constructing and solving the normal equations. $$A=\left[\begin{array}{rr} 3 & -2 \\ 1 & -2 \\ 2 & 1 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]$$

Find an SVD of the indicated matrix. $$A=\left[\begin{array}{lll}1 & 1 & 1 \\\1 & 1 & 1\end{array}\right]$$

In Exercises 19 and \(20,\langle\mathbf{u}, \mathbf{v}\rangle\) defines an inner product on \(\mathbb{R}^{2}\) where \(\mathbf{u}=\left[\begin{array}{l}u_{1} \\\ u_{2}\end{array}\right]\) and \(\mathbf{v}=\left[\begin{array}{l}v_{1} \\\ v_{2}\end{array}\right] .\) Find a symmetric matrix \(A\) \(\operatorname{such} \operatorname{that}\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}^{T} A \mathbf{v}\) $$\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{1} v_{2}+2 u_{2} v_{1}+5 u_{2} v_{2}$$

In Exercises 21 and 22 , sketch the unit circle in \(\mathbb{R}^{2}\) for the given inner product, where \(\mathbf{u}=\left[\begin{array}{l}u_{1} \\\ u_{2}\end{array}\right]\) and \(\mathbf{v}=\left[\begin{array}{l}v_{1} \\\ v_{2}\end{array}\right]\) $$\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+\frac{1}{4} u_{2} v_{2}$$

Find the third-order Fourier approximation to fon \([-\pi, \pi].\) $$f(x)=|x|$$

Find a least squares solution of \(A \mathbf{x}=\mathbf{b}\) by constructing and solving the normal equations. $$A=\left[\begin{array}{rr} 1 & -2 \\ 0 & -3 \\ 2 & 5 \\ 3 & 0 \end{array}\right], \mathbf{b}=\left[\begin{array}{r} 4 \\ 1 \\ -2 \\ 4 \end{array}\right]$$

Find a least squares solution of \(A \mathbf{x}=\mathbf{b}\) by constructing and solving the normal equations. $$A=\left[\begin{array}{rr} 2 & 0 \\ 1 & -1 \\ 3 & 1 \\ -1 & 2 \end{array}\right], \mathbf{b}=\left[\begin{array}{r} 5 \\ 1 \\ -1 \\ 3 \end{array}\right]$$

Find the third-order Fourier approximation to fon \([-\pi, \pi].\) $$f(x)=x^{2}$$

In Exercises 21 and 22 , sketch the unit circle in \(\mathbb{R}^{2}\) for the given inner product, where \(\mathbf{u}=\left[\begin{array}{l}u_{1} \\\ u_{2}\end{array}\right]\) and \(\mathbf{v}=\left[\begin{array}{l}v_{1} \\\ v_{2}\end{array}\right]\) $$\langle\mathbf{u}, \mathbf{v}\rangle=4 u_{1} v_{1}+u_{1} v_{2}+u_{2} v_{1}+4 u_{2} v_{2}$$