91Ó°ÊÓ

Verify that the given matrix is orthogonal, and find its inverse. $$(1 / \sqrt{2})\left[\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right]$$

Verify that the generating set of the given subspace \(W\) is orthogonal, and find the projection of the given vector b on \(W\). $$ W=\operatorname{sp}([2,3,1],[-1,1,-1]) ; \mathbf{b}=[2,1,4] $$

Find the projection matrix for the given subspace, and find the projection of the indicated vector on the subspace. $$ [1,2,1] \text { on } \operatorname{sp}([2,1,-1]) \text { in } \mathbb{R}^{3} $$

Find the indicated projection. The projection of \([2,1]\) on \(\mathrm{sp}([3,4])\) in \(\mathbb{R}^{2}\)

Find the projection matrix for the given subspace, and find the projection of the indicated vector on the subspace. $$ [1,3,4] \text { on sp }([1,-1,2]) \text { in } \mathbb{R}^{3} $$

A company had profits (in units of $$\$ 10,000$$ ) of \(0.5\) in 1989,1 in 1991 , and 2 in 1994 . Let time \(t\) be measured in years, with \(t=0\) in \(1989 .\) a. Find the least-squares linear fit of the data. b. Using the answer to part (a), estimate the profit in \(1995 .\)

Verify that the given matrix is orthogonal, and find its inverse. $$\left[\begin{array}{rrr}\frac{3}{5} & 0 & \frac{4}{5} \\ -\frac{4}{5} & 0 & \frac{1}{5} \\ 0 & 1 & 0\end{array}\right]$$

Verify that the generating set of the given subspace \(W\) is orthogonal, and find the projection of the given vector b on \(W\). $$ W=\operatorname{sp}([-1,0,1],[1,1,1]) ; b=[1,2,3] $$

Find the indicated projection. The projection of \([3,4]\) on \(\operatorname{sp}([2,1])\) in \(\mathbb{R}^{2}\)

Verify that the generating set of the given subspace \(W\) is orthogonal, and find the projection of the given vector b on \(W\). $$ \begin{aligned} &W=\operatorname{sp}([1,-1,-1,1],[1,1,1,1] \\ &[-1,0,0,1]) ; \mathbf{b}=[2,1,3,1] \end{aligned} $$