91Ó°ÊÓ

For each of the following matrices, determine a basis for each of the subspaces \(R\left(A^{T}\right), N(A), R(A)\) and \(N\left(A^{T}\right)\) (a) \(A=\left(\begin{array}{ll}3 & 4 \\ 6 & 8\end{array}\right)\) (b) \(A=\left(\begin{array}{lll}1 & 3 & 1 \\ 2 & 4 & 0\end{array}\right)\) (c) \(A=\left(\begin{array}{rr}4 & -2 \\ 1 & 3 \\ 2 & 1 \\ 3 & 4\end{array}\right)\) (d) \(A=\left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 1 & 1 & 2 & 2\end{array}\right)\)

Let \(\mathbf{x}=(1,1,1,1)^{T}\) and \(\mathbf{y}=(8,2,2,0)^{T}\) (a) Determine the angle \(\theta\) between \(\mathbf{x}\) and \(\mathbf{y}\) (b) Find the vector projection \(\mathbf{p}\) of \(\mathbf{x}\) onto \(\mathbf{y}\) (c) Verify that \(\mathbf{x}-\mathbf{p}\) is orthogonal to \(\mathbf{p}\) (d) Compute \(\|\mathbf{x}-\mathbf{p}\|_{2},\|\mathbf{p}\|_{2},\|\mathbf{x}\|_{2}\) and verify that the Pythagorean law is satisfied.

Let \(S\) be the subspace of \(\mathbb{R}^{3}\) spanned by \(\mathbf{x}=\) \((1,-1,1)^{T}\) (a) Find a basis for \(S^{\perp}\) (b) Give a geometrical description of \(S\) and \(S^{\perp}\)

(a) Let \(S\) be the subspace of \(\mathbb{R}^{3}\) spanned by the vectors \(\mathbf{x}=\left(x_{1}, x_{2}, x_{3}\right)^{T}\) and \(\mathbf{y}=\left(y_{1}, y_{2}, y_{3}\right)^{T}\) Let $$A=\left(\begin{array}{lll} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{array}\right)$$ Show that \(S^{\perp}=N(A)\) (b) Find the orthogonal complement of the subspace of \(\mathbb{R}^{3}\) spanned by \((1,2,1)^{T}\) and \((1,-1,2)^{T}\)

Consider the vector space \(C[-1,1]\) with inner product defined by $$\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x$$ Find an orthonormal basis for the subspace spanned by \(1, x,\) and \(x^{2}\)

Let \(A\) be a \(3 \times 2\) matrix with rank \(2 .\) Give geometric descriptions of \(R(A)\) and \(N\left(A^{T}\right),\) and describe geometrically how the subspaces are related.

Let $$A=\left(\begin{array}{ll} 2 & 1 \\ 1 & 1 \\ 2 & 1 \end{array}\right) \quad \text { and } \quad \mathbf{b}=\left(\begin{array}{r} 12 \\ 6 \\ 18 \end{array}\right)$$ (a) Use the Gram-Schmidt process to find an orthonormal basis for the column space of \(A\) (b) Factor \(A\) into a product \(Q R,\) where \(Q\) has an orthonormal set of column vectors and \(R\) is upper triangular. (c) Solve the least squares problem \(A \mathbf{x}=\mathbf{b}\)

(a) Find the best least squares fit by a linear function to the data $$\begin{array}{c|r|r|r|r} x & -1 & 0 & 1 & 2 \\ \hline y & 0 & 1 & 3 & 9 \end{array}$$ (b) Plot your linear function from part (a) along with the data on a coordinate system.

Is it possible for a matrix to have the vector (3,1,2) in its row space and \((2,1,1)^{T}\) in its null space? Explain.

If \(A\) is an \(m \times n\) matrix of rank \(r,\) what are the dimensions of \(N(A)\) and \(N\left(A^{T}\right) ?\) Explain.