91Ó°ÊÓ

a. Let \(w(t)\) be a positive-valued function in \(C[a, b]\) where \(b>a\). Verify that the rule \(\langle f, g\rangle=\) \(\int_{a}^{b} w(t) f(t) g(t) d t\) defines an inner product on \(C[a, b]\) b. If we chose the weight function \(w(t)\) so that \(\int_{a}^{b} w(t) d t=1,\) what is the norm of the constant function \(f(t)=1\) in this inner product space?

Find an orthonormal basis of the image of the matrix \\[A=\left[\begin{array}{rrr}1 & 2 & 1 \\\2 & 1 & 1 \\\2 & -2 & 0\end{array}\right]\\]

Find an orthogonal transformation \(T\) from \(\mathbb{R}^{3}\) to \(\mathbb{R}^{3}\) such that \\[ T\left[\begin{array}{l} 2 / 3 \\ 2 / 3 \\ 1 / 3 \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right] \\]

Consider an \(n \times m\) matrix \(A\) with \(\operatorname{rank}(A)=m .\) Is it always possible to write \(A\) as \\[A=Q L\\] where \(Q\) is an \(n \times m\) matrix with orthonormal columns and \(L\) is a lower triangular \(m \times m\) matrix with positive diagonal entries? Explain.

a. Find all \(n \times n\) matrices that are both orthogonal and upper triangular, with positive diagonal entries. b. Show that the \(Q R\) factorization of an invertible \(n \times n\) matrix is unique. Hint: If \(A=Q_{1} R_{1}=Q_{2} R_{2}\), then the matrix \(Q_{2}^{-1} Q_{1}=R_{2} R_{1}^{-1}\) is both orthogonal and upper triangular, with positive diagonal entries.

Find a basis of the space \(V\) of all symmetric \(3 \times 3 \mathrm{ma}\) trices, and thus determine the dimension of \(V\).

Find a basis of the space \(V\) of all skew-symmetric \(3 \times 3\) matrices, and thus determine the dimension of \(V\).

Find the dimension of the space of all skew-symmetric \(n \times n\) matrices.

Find the image and kernel of the linear transformation \(L(A)=\frac{1}{2}\left(A-A^{T}\right)\) from \(\mathbb{R}^{n \times n}\) to \(\mathbb{R}^{n \times n}\). Hint: Think about symmetric and skew-symmetric matrices.

Consider a subspace \(V\) of \(\mathbb{R}^{n}\) with a basis \(\vec{v}_{1}, \ldots, \vec{v}_{m}\) suppose we wish to find a formula for the orthogonal projection onto \(V .\) Using the methods we have developed thus far, we can proceed in two steps: We use the Gram-Schmidt process to construct an orthonormal basis \(\vec{u}_{1}, \ldots, \vec{u}_{m}\) of \(V,\) and then we use Theorem 5.3 .10: The matrix of the orthogonal projection is \(Q Q^{T},\) where \\[ Q=\left[\begin{array}{lll} \vec{u}_{1} & \dots & \vec{u}_{m} \end{array}\right] \\] In this exercise we will see how we can write the matrix of the projection directly in terms of the basis \(\vec{v}_{1}, \ldots, \vec{v}_{m}\) and the matrix \\[ A=\left[\begin{array}{lll} \vec{v}_{1} & \dots & \vec{v}_{m} \end{array}\right] \\] (This issue will be discussed more thoroughly in Section \(5.4 ;\) see Theorem \(5.4 .7 .\) since proj \(_{V} \vec{x}\) is in \(V,\) we can write \\[ \operatorname{proj}_{V} \vec{x}=c_{1} \vec{v}_{1}+\cdots+c_{m} \vec{v}_{m} \\] for some scalars \(c_{1}, \ldots, c_{m}\) yet to be determined. Now \(\vec{x}-\operatorname{proj}_{V}(\vec{x})=\vec{x}-c_{1} \vec{v}_{1}-\cdots- c_{m} \vec{v}_{m}\) is orthogonal to \(V,\) meaning that \(\vec{v}_{i} \cdot\left(\vec{x}-c_{1} \vec{v}_{1}-\cdots- c_{m} \vec{v}_{m}\right)=0\) for \(i=1, \ldots, m\) a. Use the equation \(\vec{v}_{i} \cdot\left(\vec{x}-c_{1} \vec{v}_{1}-\cdots- c_{m} \vec{v}_{m}\right)=0\) to show that \(A^{T} A \vec{c}=A^{T} \vec{x},\) where \(\vec{c}=\left[\begin{array}{c}c_{1} \\ \vdots \\ c_{m}\end{array}\right]\) b. Conclude that \(\vec{c}=\left(A^{T} A\right)^{-1} A^{T} \vec{x}\) and proj \(_{V} \vec{x}=\) \(A \vec{c}=A\left(A^{T} A\right)^{-1} A^{T} \vec{x}\).