91Ó°ÊÓ

Let \(\mathbf{u}\) be a unit vector in \(\mathbb{R}^{n}\) and let \(H=I-2 \mathbf{u u}^{T}\) Show that \(H\) is both orthogonal and symmetric and hence is its own inverse.

Let \(U\) and \(V\) be subspaces of a vector space \(W\) Show that if \(W=U \oplus V\), then \(U \cap V=\\{\mathbf{0}\\}\)

Let \(Q\) be an orthogonal matrix and let \(d=\operatorname{det}(Q)\) Show that \(|d|=1\)

Show that the product of two orthogonal matrices is also an orthogonal matrix. Is the product of two permutation matrices a permutation matrix? Explain.

How many \(n \times n\) permutation matrices are there?

Use mathematical induction to show that if \(Q \in\) \(\mathbb{R}^{n \times n}\) is both upper triangular and orthogonal, then \(\mathbf{q}_{j}=\pm \mathbf{e}_{j}, j=1, \ldots, n\)

The result of Exercise 26 is not valid for norms other than the norm derived from the inner product. Give an example of this in \(\mathbb{R}^{2}\) using \(\|\cdot\|_{1}\)

Let \(\mathbf{v}\) be a vector in an inner product space \(V\) and let \(\mathbf{p}\) be the projection of \(\mathbf{v}\) onto an \(n\) -dimensional subspace \(S\) of \(V\). Show that \(\|\mathbf{p}\| \leq\|\mathbf{v}\| .\) Under what conditions does equality occur.

Consider the inner product space \(C[0,1]\) with inner product defined by $$\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x$$ Let \(S\) be the subspace spanned by the vectors 1 and \(2 x-1\) (a) Show that 1 and \(2 x-1\) are orthogonal. (b) Determine \|1\| and \(\|2 x-1\|\) (c) Find the best least squares approximation to \(\sqrt{x}\) by a function from the subspace \(S\)

The trace of an \(n \times n\) matrix \(C,\) denoted \(\operatorname{tr}(C)\), is the sum of its diagonal entries; that is \\[ \operatorname{tr}(C)=c_{11}+c_{22}+\cdots+c_{n n} \\] If \(A\) and \(B\) are \(m \times n\) matrices, show that (a) \(\|A\|_{F}^{2}=\operatorname{tr}\left(A^{T} A\right)\) (b) \(\|A+B\|_{F}^{2}=\|A\|_{F}^{2}+2 \operatorname{tr}\left(A^{T} B\right)+\|B\|_{F}^{2}\)