91Ó°ÊÓ

Determine whether the given set of vectors is an orthogonal set in \(\mathbb{R}^{n} .\) For those that are, determine a corresponding orthonormal set of vectors. $$\\{(1,2),(-4,2)\\}$$

Use the standard inner product in \(\mathbb{R}^{5}\) to determine the angle between the vectors \(\mathbf{v}=(0,-2,1,4,1)\) and \(\mathbf{w}=(-3,1,-1,0,3)\)

Determine whether the given set of vectors is an orthogonal set in \(\mathbb{R}^{n} .\) For those that are, determine a corresponding orthonormal set of vectors. $$\\{(2,-1,1),(1,1,-1),(0,1,1)\\}$$

Determine whether the given set of vectors is an orthogonal set in \(\mathbb{R}^{n} .\) For those that are, determine a corresponding orthonormal set of vectors. $$\begin{array}{l} \\{(1,2,-1,0),(1,0,1,2),(-1,1,1,0), \\ (1,-1,-1,0)\\} \end{array}$$

Find an orthonormal basis for the row space, column space, and null space of the given matrix \(A\). $$A=\left[\begin{array}{lll} 1 & 2 & 6 \\ 2 & 1 & 6 \\ 0 & 1 & 2 \\ 1 & 0 & 2 \end{array}\right]$$

Find the equation of the least squares line associated with the given set of data points. (-7,3),(-4,0),(2,-1),(3,6),(6,-1).

Find an orthogonal basis for the span of the set \(S\) in the vector space \(V\). \(V=\mathbb{R}^{3}, S=\\{(6,-3,2),(1,1,1),(1,-8,-1)\\}.\)

Use the inner product \((5.1 .13)\) in Problem 11 to determine \(\langle A, B\rangle,\|A\|,\) and \(\|B\| .\) Also, determine the angle between the given matrices. $$A=\left[\begin{array}{rr} 2 & -1 \\ 3 & 5 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & 1 \\ -1 & 2 \end{array}\right]$$

Show that the given functions are orthonormal on [-1,1]. $$\begin{array}{l} f_{1}(x)=\cos \pi x, f_{2}(x)=\cos 2 \pi x \\ f_{3}(x)=\cos 3 \pi x \end{array}$$

Find the distance from the given point \(P\) to the given line \(L\). \(P(1,-1) ;\) Line \(L\) with equation \(4 x+5 y=1\)