魅影直播

Question: In Exercises 1 and 2, you may assume that\(\left\{ {{{\bf{u}}_{\bf{1}}},...,{{\bf{u}}_{\bf{4}}}} \right\}\)is an orthogonal basis for\({\mathbb{R}^{\bf{4}}}\).

1.\({{\bf{u}}_{\bf{1}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\\{ - {\bf{1}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{2}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{5}}\\{\bf{1}}\\{\bf{1}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{3}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{\bf{1}}\\{ - {\bf{4}}}\end{aligned}} \right]\),\({{\bf{u}}_{\bf{4}}} = \left[ {\begin{aligned}{*{20}{c}}{\bf{5}}\\{ - {\bf{3}}}\\{ - {\bf{1}}}\\{\bf{1}}\end{aligned}} \right]\),\({\bf{x}} = \left[ {\begin{aligned}{*{20}{c}}{{\bf{10}}}\\{ - {\bf{8}}}\\{\bf{2}}\\{\bf{0}}\end{aligned}} \right]\)

Write x as the sum of two vectors, one in\({\bf{Span}}\left\{ {{{\bf{u}}_1},{{\bf{u}}_2},{{\bf{u}}_3}} \right\}\)and the other in\({\bf{Span}}\left\{ {{{\bf{u}}_{\bf{4}}}} \right\}\).

Consider a consistent system $\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$.

(a) Show that this system has a solution ${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}$ in ${\left(kerA\right)}^{\mathbf{鈯�}}$ .

(b) Show that the system$\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ has only one solution in ${\left(kerA\right)}^{\mathbf{鈯�}}$ .

(c) If${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}$ is the solution in ${\mathbf{\left(}\mathbf{kerA}\mathbf{\right)}}^{\mathbf{鈯�}}$ androle="math" localid="1660124695419" ${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_1$is another solution of the system $\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ , show that$||{\stackrel{鈫�}{x}}_0||\mathbf{<}||{\stackrel{鈫�}{x}}_1||$ . The vector${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}_{\mathbf{0}}$ is called the minimal solution of the linear system $\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$ .

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

15.$\left[\begin{array}{c}2\\ 1\\ -2\end{array}\right]$

Consider the subspace$\mathit{i}\mathit{m}\left(A\right)$ of ${\mathbf{鈩�}}^{\mathbf{2}}$ . Where $\mathit{A}\mathbf{=}\left[\begin{array}{cc}2& 4\\ 3& 6\end{array}\right]$ . Find a basis of $\mathit{k}\mathit{e}\mathit{r}\left(A^T\right)$ , and draw a sketch illustrating the formula$\left(im\left(A^T\right)\right)\mathbf{=}\mathit{k}\mathit{e}\mathit{r}\left(A^T\right)$in this case.

By using paper and pencil, find the least squares${\stackrel{\mathbf{鈫�}}{\mathbf{x}}}^{\mathbf{*}}$ of the system $\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{b}}$, where$\mathit{A}\mathbf{=}\left[\begin{array}{cc}1& 1\\ 1& 0\\ 1& 1\end{array}\right]$ and$\stackrel{\mathbf{鈫�}}{\mathbf{b}}\mathbf{=}\left[\begin{array}{c}3\\ 3\\ 3\end{array}\right]$. Verify that the vector$\stackrel{\mathbf{鈫�}}{\mathbf{b}}\mathbf{-}\mathit{A}{\stackrel{\mathbf{鈫�}}{\mathbf{x}}}^{\mathbf{*}}$ is perpendicular to the image of A.

(a) Consider an matrix A such that ${\mathit{A}}^{\mathbf{螕}}\mathit{A}\mathbf{=}{\mathit{I}}_{\mathbf{m}}$. It is necessarily true that? Explain.

(b) Consider an $\mathit{n}\mathbf{脳}\mathit{n}$matrix A such that ${\mathit{A}}^{\mathbf{T}}\mathit{A}\mathbf{=}{\mathit{I}}_{\mathbf{n}}$. Is it necessarily true that $\mathit{A}{\mathit{A}}^{\mathbf{T}}\mathbf{=}{\mathit{I}}_{\mathbf{n}}$? Explain.

Find an orthonormal basis of the kernel of the matrix $\mathbf{A}\mathbf{=}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& 2& 3& 4\end{array}\right]$.

Let S(t)be the number of daylight hours on the tth day of the year 2012 in Rome, Italy. We are given the following data for S(t):

We wish to fit a trigonometric function of the form

$\mathbf{f}\left(t\right)\mathbf{=}\mathbf{a}\mathbf{+}\mathbf{bsin}\left(\frac{2\mathrm{蟿蟿}}{366}t\right)\mathbf{+}\mathbf{c}\mathbf{}\mathbf{cos}\left(\frac{2\mathrm{蟿蟿}}{366}t\right)$

To these data. Find the best approximation of this form, using least squares. How many daylight hours does your model predict for the longest day of the year 2012? (The actual value is 15 hours, 13 minutes, 39 seconds.)

Consider an n x m matrix A with rank (A) = m. Is it always possible to write A as A = QL where Q is an n x m matrix with orthonormal columns and L is a lower triangular m x m matrix with positive diagonal entries? Explain.

Let Abe an$\mathbf{n}\mathbf{脳}\mathbf{m}$matrix. Is the formula${\left(\mathrm{kerA}\right)}^{\mathbf{鈯�}}\mathbf{=}\mathbf{im}\left(A^T\right)$necessarily true? Explain.