91Ó°ÊÓ

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). $$f(x)=2 \sin x \cos x, \text { fourth order }$$

Let \(A\) be an \(m \times n\) matrix. (a) Explain why \(R\left(A^{T}\right)\) is the same as the row space of \(A\). (b) Prove that \(N(A) \subset R\left(A^{T}\right)^{\perp}\). (c) Prove that \(N(A)=R\left(A^{T}\right)^{\perp}\). (d) Prove that \(N\left(A^{T}\right)=R(A)^{\perp}\).

Find the orthogonal projection of \(f\) onto \(g\). Use the inner product in \(C[a, b]\) \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\). \(C[0,1], \quad f(x)=x, \quad g(x)=e^{-x}\)

Determine all vectors \(\mathbf{v}\) that are orthogonal to \(\mathbf{u}\). \(\mathbf{u}=(2,7)\)

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

Determine all vectors \(\mathbf{v}\) that are orthogonal to \(\mathbf{u}\). \(\mathbf{u}=(-3,2)\)

Find the orthogonal projection of \(f\) onto \(g\). Use the inner product in \(C[a, b]\) \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\). \(C[-\pi, \pi], \quad f(x)=\sin x, \quad g(x)=\cos x\)

Find the orthogonal projection of \(f\) onto \(g\). Use the inner product in \(C[a, b]\) \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\). \(C[-\pi, \pi], \quad f(x)=\sin 2 x, \quad g(x)=\cos 2 x\)

Determine all vectors \(\mathbf{v}\) that are orthogonal to \(\mathbf{u}\). \(\mathbf{u}=(4,-1,0)\)

Find the orthogonal projection of \(f\) onto \(g\). Use the inner product in \(C[a, b]\) \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\). \(C[-\pi, \pi], \quad f(x)=x, \quad g(x)=\sin 2 x\)