91Ó°ÊÓ

Suppose that \(A\) is an \(m \times n\) matrix in which no two columns are identical. Prove that \(A^{*} A\) is a diagonal matrix if and only if every pair of columns of \(A\) is orthogonal.

For each of the sets of data that follows, use the least squares approximation to find the best fits with both (i) a linear function and (ii) a quadratic function. Compute the error \(E\) in both cases. (a) \(\\{(-3,9),(-2,6),(0,2),(1,1)\\}\) (b) \(\\{(1,2),(3,4),(5,7),(7,9),(9,12)\\}\) (c) \(\\{(-2,4),(-1,3),(0,1),(1,-1),(2,-3)\\}\)

Let \(A\) be a square matrix such that \(A^{2}=O .\) Prove that \(\left(A^{\dagger}\right)^{2}=O .\)

Let \(V\) be a finite-dimensional inner product space, and let \(T\) and \(U\) be self-adjoint operators on \(V\) such that \(T\) is positive definite. Prove that both TU and UT are diagonalizable linear operators that have only real eigenvalues. Hint: Show that UT is self-adjoint with respect to the inner product \(\langle x, y\rangle^{\prime}=\langle\mathrm{T}(x), y\rangle\). To show that TU is self-adjoint, repeat the argument with \(\mathrm{T}^{-1}\) in place of \(\mathrm{T}\).

Let \(V=C([-1,1])\) with the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(t) g(t) d t\), and let \(\mathrm{W}\) be the subspace \(\mathrm{P}_{2}(R)\), viewed as a space of functions. Use the orthonormal basis obtained in Example 5 to compute the "best" (closest) second-degree polynomial approximation of the function \(h(t)=\) \(e^{t}\) on the interval \([-1,1]\).

Let \(V\) be a complex inner product space with an inner product \(\langle\cdot \cdot \cdot\rangle\). Let \([\cdot, \cdot]\) be the real-valued function such that \([x, y]\) is the real part of the complex number \(\langle x, y\rangle\) for all \(x, y \in \mathrm{V}\). Prove that \([\cdot, \cdot \cdot]\) is an inner product for \(\mathrm{V}\), where \(\mathrm{V}\) is regarded as a vector space over \(R\). Prove, furthermore, that \([x, i x]=0\) for all \(x \in \mathrm{V}\).

Find new coordinates \(x^{\prime}, y^{\prime}\) so that the following quadratic forms can be written as \(\lambda_{1}\left(x^{\prime}\right)^{2}+\lambda_{2}\left(y^{\prime}\right)^{2}\). (a) \(x^{2}+4 x y+y^{2}\) (b) \(2 x^{2}+2 x y+2 y^{2}\) (c) \(x^{2}-12 x y-4 y^{2}\) (d) \(3 x^{2}+2 x y+3 y^{2}\) (e) \(x^{2}-2 x y+y^{2}\)

Let \(V\) and \(W\) be finite-dimensional inner product spaces, and let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) be a linear transformation. Prove part (b) of the lemma to Theorem 6.30: \(\mathrm{TT}^{\dagger}\) is the orthogonal projection of \(\mathrm{W}\) on \(\mathrm{R}(\mathrm{T})\).