91Ó°ÊÓ

In Exercise 40 through 46, consider vectors ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{3}}$in ${\mathit{R}}^{\mathbf{4}}$; we are told that ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{i}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{j}}$is the entry ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$of matrix A.

role="math" localid="1659603341436" $\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$

Find $\mathbf{\left|}\mathbf{\right|}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{\left|}\mathbf{\right|}$

Consider an invertible n x n matrix A whose columns are orthogonal, but not necessarily orthonormal. What does the QR factorization of A look like?

Consider the data in the following table.

Use the methods discussed in Exercise 39 to fit a power function of the form$\mathit{D}\mathbf{=}\mathit{k}{\mathit{a}}^{\mathbf{n}}$ to these data. Explain, in terms of Kepler’s laws of planetary motion. Explain why the constant kis close to 1.

TRUE OR FALSE

If A is a square matrix such that${\mathit{A}}^{\mathbf{T}}\mathit{A}\mathbf{=}\mathit{A}{\mathit{A}}^{\mathbf{T}}$ then $\mathit{k}\mathit{e}\mathit{r}\left(A\right)\mathbf{=}\mathit{k}\mathit{e}\mathit{r}\left(A^T\right)$.

Find the matrix Aof the orthogonal projection onto the line in${\mathit{R}}^{\mathbf{n}}$spanned by the vector

$\begin{array}{r}\left[\begin{array}{c}1\\ 1\\ .\\ .\\ .\\ 1\end{array}\right]\\ \end{array}\}$all ncomponents are 1.

TRUE OR FALSE

41. Determine whether the statement “Any square matrix can be written as the sum of the symmetric and a skew-symmetric matrix.

In Exercise 40 through 46, consider vectors ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{3}}$in ${\mathit{R}}^{\mathbf{4}}$; we are told that ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{i}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{j}}$is the entry ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$ of matrix A.

$\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$

Find the angle enclosed by vectors ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}$and ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{3}}$

Consider an invertible upper triangular n x n, matrix A. What does the QR factorization of A look like?

If Ais any matrix, show that the linear transformation $\mathit{L}\mathbf{(}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{→}}{\mathbf{x}}$from $\mathbf{im}\mathbf{\left(}{\mathbf{A}}^{\mathbf{T}}\mathbf{\right)}$to $\mathbf{im}\mathbf{\left(}\mathbf{A}\mathbf{\right)}$is an isomorphism. This provides yet another proof of the formula $\mathbf{rank}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{rank}\mathbf{\left(}{\mathbf{A}}^{\mathbf{T}}\mathbf{\right)}$.

In Exercise 40 through 46, consider vectors ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{3}}$in role="math" localid="1659606624550" ${\mathit{R}}^{\mathbf{4}}$; we are told that ${\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{i}}\mathbf{,}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{j}}$is the entry data-custom-editor="chemistry" ${\mathit{a}}_{\mathbf{i}\mathbf{j}}$of matrix A.

$\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}3& 5& 11\\ 5& 9& 20\\ 11& 20& 49\end{array}\right]$

Find $\mathbf{\left|}\mathbf{\right|}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{+}{\stackrel{\mathbf{→}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{\left|}\mathbf{\right|}$