91影视

Consider a symmetric $\mathbf{n}\mathbf{脳}\mathbf{m}$matrix A. What is the relationship between Im(A)and ker(A)?

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

7.$\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}-2\\ 1\\ 2\end{array}\right]\mathbf{,}\left[\begin{array}{c}18\\ 0\\ 0\end{array}\right]$

If the$\mathbf{n}\mathbf{脳}\mathbf{n}$matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?AB.

All nonzero symmetric matrices are invertible.

Question:All nonzero symmetric matrices are invertible.

If the$\mathbf{n}\mathbf{脳}\mathbf{n}$ matrices Aand Bare orthogonal, which of the matrices in Exercise 5 through 11 must be orthogonal as well?A+B.

Consider a linear transformation $\mathit{L}\left(\stackrel{鈬赌}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$fromlocalid="1662031612255">鈩�nto localid="1662031573789">鈩�m, with $\mathit{k}\mathit{e}\mathit{r}\left(L\right)\mathbf{=}\left\{\stackrel{鈬赌}{0}\right\}$. The pseudoinverse L⁺ of L is the transformation fromlocalid="1662031665028">鈩�mto localid="1662031688923">鈩�ngiven by ${\mathit{L}}^{\mathbf{+}}\left(y\right)\mathbf{=}\left(theleast-squaressolutionofL\left(\stackrel{鈬赌}{x}\right)=\stackrel{鈬赌}{y}\right)$.

1. Show that the transformation L⁺ is linear. Find the matrix A⁺ of L⁺ , in terms of the matrix A of L.
2. If L is invertible, what is the relationship between L⁺ and L^-1 ?
3. What is localid="1660646282368" ${\mathit{L}}^{\mathbf{+}}\mathbf{\left(}\mathbf{L}\left(\stackrel{鈬赌}{x}\right)\mathbf{\right)}$, for $\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$in localid="1662031720578">鈩�n?
4. What is ${\mathit{L}}^{\mathbf{+}}\mathbf{\left(}\mathbf{L}\mathbf{\left(}\stackrel{\mathbf{鈬赌}}{\mathbf{y}}\mathbf{\right)}\mathbf{\right)}$, for$\stackrel{\mathbf{鈬赌}}{\mathbf{y}}$in localid="1662031731206" ${\mathbf{鈩�}}^{\mathbf{m}}$?
5. Find L⁺ for the linear transformation $\mathit{L}\left(\stackrel{鈬赌}{x}\right)\mathbf{=}\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\stackrel{\mathbf{鈬赌}}{\mathbf{x}}$.

For each pair of vectors $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$and $\stackrel{\mathbf{鈫�}}{\mathbf{v}}$listed in Exercises 7 through 9, determine whether the angle $\mathit{胃}$between $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$and $\stackrel{\mathbf{鈫�}}{\mathbf{v}}$is acute, obtuse, or right.

8.$\stackrel{\mathbf{鈫�}}{\mathbf{u}}\mathbf{=}\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]\mathbf{,}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\left[\begin{array}{c}2\\ -8\\ 5\end{array}\right]$

If A is an $\mathbf{n}\mathbf{脳}\mathbf{n}$matrix such that role="math" localid="1659514225617" ${\mathbf{AA}}^{\mathbf{T}}\mathbf{=}{\mathbf{I}}_{\mathbf{n}}$, then A must be an orthogonal matrix.

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

8.$\left[\begin{array}{c}5\\ 4\\ 2\\ 2\end{array}\right]\mathbf{,}\left[\begin{array}{c}3\\ 6\\ 7\\ -2\end{array}\right]$