91Ó°ÊÓ

Does the equation$\mathit{r}\mathit{a}\mathit{n}\mathit{k}\left(A\right)\mathbf{=}\mathit{r}\mathit{a}\mathit{n}\mathit{k}\left(A^{T}A\right)$ holds for all$\mathit{n}\mathbf{×}\mathit{m}$ matrices A? Explain.

Using paper and pencil, find the QR factorization of the matrices in Exercises 15 through 28. Compare with Exercises 1 through 14.

17.$\left[\begin{array}{cc}4& 25\\ 0& 0\\ 3& -25\end{array}\right]$

Consider a linear space V. For which linear transformations Tfrom Vto is

$$\begin{gathered} {\mathit{R}}^{\mathbf{n}}\mathbf{}\mathit{i}\mathit{s} \\ <v,w>\mathbf{=}\mathit{T}\left(v\right)\mathbf{.}\mathit{T}\left(w\right) \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{↑}\mathbf{}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{D}\mathit{o}\mathit{t}\mathbf{}\mathit{p}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t} \end{gathered}$$

an inner product in V?

Find a basis for${\mathit{W}}^{\mathbf{âŠ\yen }}$, where$\mathit{W}\mathbf{=}\mathit{s}\mathit{p}\mathit{a}\mathit{n}\left\{\left[\begin{array}{c}1\\ 2\\ 3\\ 4\end{array}\right],\left[\begin{array}{c}5\\ 6\\ 7\\ 8\end{array}\right]\right\}$.

Question: If the $\mathbf{n}\mathbf{×}\mathbf{n}$matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?role="math" localid="1659492178067" ${\mathbf{B}}^{\mathbf{-}\mathbf{1}}$.

If Aand Bare symmetric n×nmatrices, then A B B Amust be symmetric as well.

If matrices Aand Bcommute, then matrices ${\mathit{A}}^{\mathbf{T}}$and ${\mathit{B}}^{\mathbf{T}}$must commute as well.

Consider an orthonormal basis B of the inner product space V. For an element fof V, what is the relationship between $||f||$and $||{\left[f\right]}_B||$(the norm in ${\mathbf{R}}^{\mathbf{n}}$defined by the dot product)?

Here is an infinite-dimensional version of Euclidean space: In the space of all infinite sequences, consider the subspace ${\mathit{l}}_{\mathbf{2}}$of square-summable sequences [i.e., those sequences $\left(x_1,x_2,....\right)$for which the infinite series ${\mathit{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathit{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}\mathit{L}$converges]. For $\stackrel{\mathbf{âŠ\yen }}{\mathbf{x}}$and $\stackrel{\mathbf{âŠ\yen }}{\mathbf{y}}$in ${\mathit{l}}_{\mathbf{2}}$, we define $||\stackrel{r}{x}||\mathbf{=}\sqrt{{\mathbf{x}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}}\stackrel{\mathbf{r}}{\mathbf{x}}\mathbf{.}\stackrel{\mathbf{r}}{\mathbf{y}}\mathbf{=}{\mathit{x}}_{\mathbf{1}}{\mathit{y}}_{\mathbf{1}}\mathbf{+}{\mathit{x}}_{\mathbf{2}}{\mathit{y}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$

(Why does the series ${\mathit{x}}_{\mathbf{1}}{\mathit{y}}_{\mathbf{1}}\mathbf{+}{\mathit{x}}_{\mathbf{2}}{\mathit{y}}_{\mathbf{1}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$converge?)

a.Check that $\stackrel{\mathbf{r}}{\mathbf{x}}\mathbf{=}\left(1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},....\right)$is in ${\mathit{l}}_{\mathbf{2}}$, andfind role="math" localid="1659931721089" $||\stackrel{âŠ\yen }{x}||$. Recall the formula for the geometric series:

$\mathbf{1}\mathbf{+}\mathit{a}\mathbf{+}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{a}}^{\mathbf{3}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}\mathbf{1}\mathbf{/}\left(1-a\right)\mathbf{}\mathbf{,}\mathbf{}\mathit{i}\mathit{f}\mathbf{}\mathbf{-}\mathit{a}\mathbf{<}\mathit{a}\mathbf{<}\mathbf{1}\mathbf{.}$

b.Find the angle between $\left(1,0,0,....\right)$and role="math" localid="1659932346489" $(1,\frac{1}{2},\frac{1}{4},\frac{1}{8},....)$

c.Give an example of a sequence $\left(x_1,x_2,...\right)$that converges to 0 (i.e.,$\underset{\mathbf{n}\mathbf{→}\mathbf{∞}}{\mathbf{l}\mathbf{i}\mathbf{m}}{\mathit{x}}_{\mathbf{n}}\mathbf{=}\mathbf{0}$) but does not belong to${\mathit{l}}_{\mathbf{2}}$

d.Let Lbe the subspace of ${\mathit{l}}_{\mathbf{2}}$spanned by$\mathbf{(}\mathbf{1}\mathbf{,}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{,}\frac{\mathbf{1}}{\mathbf{8}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$ Find the orthogonal projection of (1,0,0,...) onto L.

The Hilbert space${\mathit{l}}_{\mathbf{2}}$ was initially used mostly in physics: Werner Heisenberg’s formulation of quantum mechanics is in terms of${\mathit{l}}_{\mathbf{2}}$ . Today, this space is used in many other applications, including economics. See, for example, the work of the economist Andreu Mas-Colell of the University of Barcelona.

Does the equation $\mathit{r}\mathit{a}\mathit{n}\mathit{k}\mathbf{\left(}{\mathit{A}}^{\mathbf{T}}\mathit{A}\mathbf{\right)}\mathbf{=}\mathit{r}\mathit{a}\mathit{n}\mathit{k}\mathbf{\left(}\mathit{A}{\mathit{A}}^{\mathbf{T}}\mathbf{\right)}$ holds for all$\mathit{n}\mathbf{×}\mathit{m}$ matrices? Explain.