91Ó°ÊÓ

Find bases for the four fundamental subspaces of the matrix \(A\) shown below. \(\begin{array}{ll}N(A)=\text { nullspace of } A & N\left(A^{T}\right)=\text { nullspace of } A^{T} \\ R(A)=\text { column space of } A & R\left(A^{T}\right)=\text { column space of } A^{T}\end{array}\). Then show that \(N(A)=R\left(A^{T}\right)^{\perp}\) and \(N\left(A^{T}\right)=R(A)^{\perp}\). $$\left[\begin{array}{rrr} 0 & 1 & -1 \\ 0 & -2 & 2 \\ 0 & -1 & 1 \end{array}\right]$$

Find the angle \(\theta\) between the vectors. \(\mathbf{u}=(1,-1,0,1), \quad \mathbf{v}=(-1,2,-1,0)\)

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

Find bases for the four fundamental subspaces of the matrix \(A\) shown below. \(\begin{array}{ll}N(A)=\text { nullspace of } A & N\left(A^{T}\right)=\text { nullspace of } A^{T} \\ R(A)=\text { column space of } A & R\left(A^{T}\right)=\text { column space of } A^{T}\end{array}\). Then show that \(N(A)=R\left(A^{T}\right)^{\perp}\) and \(N\left(A^{T}\right)=R(A)^{\perp}\). $$\left[\begin{array}{lll} 1 & 0 & 1 \\ 1 & 1 & 1 \end{array}\right]$$

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

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). \(f(x)=1+x,\) third order

Find bases for the four fundamental subspaces of the matrix \(A\) shown below. \(\begin{array}{ll}N(A)=\text { nullspace of } A & N\left(A^{T}\right)=\text { nullspace of } A^{T} \\ R(A)=\text { column space of } A & R\left(A^{T}\right)=\text { column space of } A^{T}\end{array}\). Then show that \(N(A)=R\left(A^{T}\right)^{\perp}\) and \(N\left(A^{T}\right)=R(A)^{\perp}\). $$\left[\begin{array}{rrrrr} 0 & 0 & 1 & 2 & 0 \\ 1 & -2 & 0 & 2 & 0 \\ -1 & 2 & 1 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 \end{array}\right]$$

Find the Fourier approximation of the specified order for the function on the interval \([0,2 \pi]\). \(f(x)=1+x, \quad\) fourth order

Find the orthogonal projection of \(f\) onto \(g\). Use the inner product in \(C[a, b]\) \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\). \(C[-1,1], \quad f(x)=x^{3}-x, \quad g(x)=2 x-1\)

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