91Ó°ÊÓ

Proof Prove that if \(\mathbf{w}\) is orthogonal to each vector in \(S=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\\},\) then \(\mathbf{w}\) is orthogonal to every linear combination of vectors in \(S\) Getting Started: To prove that \(\mathbf{w}\) is orthogonal to every linear combination of vectors in \(S,\) you need to show that their dot product is 0 (i) Write \(\mathbf{v}\) as a linear combination of vectors, with arbitrary scalars \(c_{1}, \ldots, c_{n},\) in \(S\) (ii) Form the inner product of \(\mathbf{w}\) and \(\mathbf{v}\) (iii) Use the properties of inner products to rewrite the inner product \(\langle\mathbf{w}, \mathbf{v}\rangle\) as a linear combination of the inner products \(\left\langle\mathbf{w}, \mathbf{v}_{i}\right\rangle, i=1, \ldots, n\) (iv) Use the fact that \(w\) is orthogonal to each vector in \(S\) to lead to the conclusion that \(\mathbf{w}\) is orthogonal to \(\mathbf{v}\).

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). \(f(x)=\pi-x,\) fourth order

(a) find projy \(\mathbf{u},\) (b) find proju \(\mathbf{v},\) and (c) sketch a graph of both projy \(\mathbf{u}\) and \(\operatorname{proj}_{\mathbf{u}} \mathbf{v}\). \(\mathbf{u}=(-1,3), \quad \mathbf{v}=(4,4)\)

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). \(f(x)=(x-\pi)^{2}, \quad\) third order

Let \(P\) be an \(n \times n\) matrix. Prove that the following conditions are equivalent. (a) \(P^{-1}=P^{T}\). (Such a matrix is called orthogonal.) (b) The row vectors of \(P\) form an orthonormal basis for \(R^{n}\) (c) The column vectors of \(P\) form an orthonormal basis for \(R^{n}\).

Find the angle \(\theta\) between the vectors. \(\mathbf{u}=\left(\cos \frac{\pi}{6}, \sin \frac{\pi}{6}\right), \quad \mathbf{v}=\left(\cos \frac{3 \pi}{4}, \sin \frac{3 \pi}{4}\right)\)

(a) find projy \(\mathbf{u},\) (b) find proju \(\mathbf{v},\) and (c) sketch a graph of both projy \(\mathbf{u}\) and \(\operatorname{proj}_{\mathbf{u}} \mathbf{v}\). \(\mathbf{u}=(2,-2), \quad \mathbf{v}=(3,1)\)

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). \(f(x)=(x-\pi)^{2}, \quad\) fourth order

Find the angle \(\theta\) between the vectors. \(\mathbf{u}=\left(\cos \frac{\pi}{3}, \sin \frac{\pi}{3}\right), \quad \mathbf{v}=\left(\cos \frac{\pi}{4}, \sin \frac{\pi}{4}\right)\)

Find an orthonormal basis for \(R^{4}\) that includes the vectors \(\mathbf{v}_{1}=\left(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}, 0\right) \quad\) and \(\quad \mathbf{v}_{2}=\left(0,-\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right)\).