Chapter 5

Determine the angle between the given vectors \(\mathbf{u}\) and \(\mathbf{v}\) using the standard inner product on \(\mathbb{R}^{n}\). \(\mathbf{u}=(2,3)\) and \(\mathbf{v}=(4,-1).\)

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(1,2,3),(6,-3,0)\\}$$

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 the angle between the given vectors \(\mathbf{u}\) and \(\mathbf{v}\) using the standard inner product on \(\mathbb{R}^{n}\). \(\mathbf{u}=(-2,-1,2,4)\) and \(\mathbf{v}=(-3,5,1,1).\)

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(1,-1,-1),(2,1,-1)\\}$$

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. $$\\{(2,-1,1),(1,1,-1),(0,1,1)\\}$$

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

Find the equation of the least squares line associated with the given set of data points. (1,10),(2,20),(3,10).

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(2,1,-2),(1,3,-1)\\}$$

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. $$\\{(1,3,-1,1),(-1,1,1,-1),(1,0,2,1)\\}$$

If \(f(x)=\sin x\) and \(g(x)=x\) on \([0, \pi],\) use the function inner product (5.1.5) to determine the angle between \(f\) and \(g\)

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. (2,-2),(1,-3),(0,0).

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(1,-5,-3),(0,-1,3),(-6,0,-2)\\}$$

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. $$\begin{array}{l} \\{(1,2,-1,0),(1,0,1,2),(-1,1,1,0), \\ (1,-1,-1,0)\\} \end{array}$$

If \(f(x)=\sin x\) and \(g(x)=2 \cos x+4\) on \([0, \pi / 2]\) use the function inner product ( 5.1 .5 ) to determine the angle between \(f\) and \(g\)

Find an orthonormal basis for the row space, column space, and null space of the given matrix \(A\). $$A=\left[\begin{array}{rrr} 1 & 3 & 5 \\ -1 & -3 & 1 \\ 0 & 2 & 3 \\ 1 & 5 & 2 \\ 1 & 5 & 8 \end{array}\right]$$

Find the equation of the least squares line associated with the given set of data points. (2,5),(0,-1),(5,3),(1,-3).

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(2,0,1),(-3,1,1),(1,-3,8)\\}$$

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. $$\\{(1,2,-1,0,3),(1,1,0,2,-1),(4,2,-4,-5,-4)\\}$$

Let \(m\) and \(n\) be positive real numbers. If \(f(x)=x^{m}\) and \(g(x)=x^{n}\) on an arbitrary interval \([a, b],\) use the function inner product (5.1.5) to determine the angle between \(f\) and \(g\) in terms of \(a, b, m,\) and \(n\)

Find an orthogonal basis for the span of the set \(S\) in the vector space \(V\). \(V=\mathbb{R}^{3}, S=\\{(5,-1,2),(7,1,1)\\}.\)

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

Let \(\mathbf{v}=(7,-2) .\) Determine all nonzero vectors \(\mathbf{w}\) in \(\mathbb{R}^{2}\) such that \(\\{\mathbf{v}, \mathbf{w}\\}\) is an orthogonal set.

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(-1,1,1,1),(1,2,1,2)\\}$$

If \(\mathbf{v}=(2+i, 3-2 i, 4+i)\) and \(\mathbf{w}=(-1+i, 1-3 i\) \(3-i),\) use the standard inner product in \(\mathbb{C}^{3}\) to deter\(\operatorname{mine},\langle\mathbf{v}, \mathbf{w}\rangle,\|\mathbf{v}\|,\) and \(\|\mathbf{w}\|\)

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

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).

Let \(\mathbf{v}=(-3,6,1) .\) Determine all vectors in \(\mathbb{R}^{3}\) that are orthogonal to v. Use this to find an orthogonal basis for \(\mathbb{R}^{3}\) that includes the vector \(\mathbf{v}\).

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(1,0,-1,0),(1,1,-1,0),(-1,1,0,1)\\}$$

If \(\mathbf{v}=(6-3 i, 4,-2+5 i, 3 i)\) and \(\mathbf{w}=(i, 2 i, 3 i, 4 i)\) use the standard inner product in \(\mathbb{C}^{4}\) to determine, \(\langle\mathbf{v}, \mathbf{w}\rangle,\|\mathbf{v}\|,\) and \(\|\mathbf{w}\|\)

Let \(\mathbf{v}_{1}=(1,2,3), \mathbf{v}_{2}=(1,1,-1) .\) Determine all nonzero vectors \(\mathbf{w}\) in \(\mathbb{R}^{3}\) such that \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{w}\right\\}\) is an orthogonal set. Hence obtain an orthonormal set of vectors in \(\mathbb{R}^{3}\).

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(1,2,0,1),(2,1,1,0),(1,0,2,1)\\}$$

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]\) and \(B=\left[\begin{array}{ll}b_{11} & b_{12} \\\ b_{21} & b_{22}\end{array}\right]\) be vectors in \(M_{2}(\mathbb{R}) .\) Show that the mapping $$\langle A, B\rangle=a_{11} b_{11}$$ does not define a valid inner product on \(M_{2}(\mathbb{R}) .\) Which of the four properties of an inner product, if any, do hold? Justify your answer.

Find an orthogonal basis for the span of the set \(S\) in the vector space \(V\). \(V=M_{2}(\mathbb{R}), S=\left\\{\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right],\left[\begin{array}{ll}3 & 4 \\ 4 & 3\end{array}\right],\left[\begin{array}{ll}-2 & -1 \\ -1 & -2\end{array}\right]\right.\) \(\left.\left[\begin{array}{rr}-3 & 0 \\ 0 & 3\end{array}\right],\left[\begin{array}{ll}2 & 0 \\ 0 & 0\end{array}\right]\right\\},\) using the inner product defined in Problem 11 of Section 5.1

Let \(\mathbf{v}_{1}=(-4,0,0,1), \mathbf{v}_{2}=(1,2,3,4) .\) Determine all vectors in \(\mathbb{R}^{4}\) that are orthogonal to both \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2} .\) Use this to find an orthogonal basis for \(\mathbb{R}^{4}\) that includes the vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\)

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(1,1,-1,0),(-1,0,1,1),(2,-1,2,1)\\}$$

Let \(t_{0}, t_{1}, \ldots, t_{n}\) be real numbers. For \(p\) and \(q\) in \(P_{n}(\mathbb{R}),\) define $$\langle p, q\rangle=p\left(t_{0}\right) q\left(t_{0}\right)+p\left(t_{1}\right) q\left(t_{1}\right)+\cdots+p\left(t_{n}\right) q\left(t_{n}\right)$$ (a) Prove that \(\langle,\rangle\) defines a valid inner product on \(P_{n}(\mathbb{R})\). (b) Let \(t_{0}=-3, t_{1}=-1, t_{2}=1,\) and \(t_{3}=3 .\) Let \(p_{0}(t)=1, p_{1}(t)=t,\) and \(p_{2}(t)=t^{2} .\) Show that \(p_{0}\) and \(p_{1}\) are orthogonal in this inner product space. (c) Find a polynomial \(q\) that is orthogonal to \(p_{0}\) and \(p_{1},\) such that \(\left\\{p_{0}, p_{1}, q\right\\}\) is an orthogonal basis for \(\operatorname{span}\left\\{p_{0}, p_{1}, p_{2}\right\\}\)

Show that the given set of vectors is an orthogonal set in \(\mathbb{C}^{n}\), and hence obtain an orthonormal set of vectors in \(\mathbb{C}^{n}\) in each case. $$\\{(2+i,-5 i),(-2-i,-i)\\}$$

Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors. $$\\{(1,2,3,4,5),(-7,0,1,-2,0)\\}$$

Find the distance from the point (2,3,4) to the line in \(\mathbb{R}^{3}\) passing through (0,0,0) and (6,-1,-4).

Show that the given set of vectors is an orthogonal set in \(\mathbb{C}^{n}\), and hence obtain an orthonormal set of vectors in \(\mathbb{C}^{n}\) in each case. $$\\{(1-i, 3+2 i),(2+3 i, 1-i)\\}$$

Determine orthogonal bases for rowspace( \(A\) ) and colspace( \(A\) ). $$A=\left[\begin{array}{rrrrr} 1 & -3 & 2 & 0 & -1 \\ 4 & -9 & -1 & 1 & 2 \end{array}\right]$$

Find the distance from the point \(P(0,0,0)\) to the plane with equation \(2 x-y+3 z=6\).

Determine orthogonal bases for rowspace( \(A\) ) and colspace( \(A\) ). $$A=\left[\begin{array}{ll} 1 & 5 \\ 2 & 4 \\ 3 & 3 \\ 4 & 2 \\ 5 & 1 \end{array}\right]$$

Find the distance from the point \(P(-1,3,5)\) to the plane with equation \(-x+3 y+3 z=8\).

Consider the vectors \(\mathbf{v}=(1-i, 1+2 i), \mathbf{w}=(2+i, z)\) in \(\mathrm{C}^{2}\). Determine the complex number \(z\) such that \(\\{\mathbf{v}, \mathbf{w}\\}\) is an orthogonal set of vectors, and hence obtain an orthonormal set of vectors in \(\mathbb{C}^{2}\).

Determine orthogonal bases for rowspace( \(A\) ) and colspace( \(A\) ). $$A=\left[\begin{array}{rrr} 3 & 1 & 4 \\ 1 & -2 & 1 \\ 1 & 5 & 2 \end{array}\right]$$

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. $$A=\left[\begin{array}{rr} 2 & -1 \\ 3 & 5 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & 1 \\ -1 & 2 \end{array}\right]$$