91影视

Let V be the space of all infinite sequences of real numbers. See Example 5. Which of the subsets of given in Exercises 12 through 15 are subspaces of V ? The arithmetic sequences [i.e., sequences of the form$\mathbf{(}\mathbf{a}\mathbf{,}\mathbf{a}\mathbf{+}\mathbf{k}\mathbf{,}\mathbf{a}\mathbf{+}\mathbf{2}\mathbf{k}\mathbf{,}\mathbf{a}\mathbf{+}\mathbf{3}\mathbf{k}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$ , for some constants and K .

Let Vbe the space of all infinite sequences of real numbers. See Example 5. Which of the subsets ofgiven in Exercises 12 through 15 are subspaces of V? The geometric sequences [i.e., sequences of the form$\mathbf{(}\mathbf{a}\mathbf{,}\mathbf{ar}\mathbf{,}{\mathbf{ar}}^{\mathbf{2}}\mathbf{,}{\mathbf{ar}}^{\mathbf{3}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$, for some constantsand K.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms.

13.$\mathit{T}\mathbf{\left(}\mathbf{M}\mathbf{\right)}\mathbf{=}\mathit{M}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{2}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{-}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{2}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}\mathit{M}$, from ${\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$to${\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$

In Exercises 5 through 40, find the matrix of the given linear transformation with respect to the given basis. If no basis is specified, use standard basis: $\mathit{渭}\mathbf{=}\left(1,t,t\right)\mathbf{}\mathit{f}\mathit{o}\mathit{r}\mathbf{}{\mathit{P}}_{\mathbf{2}}$

role="math" localid="1659440957055" $\mathbf{渭}\mathbf{=}\mathbf{(}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{1}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{)}$

for ${鈻�}^{2脳2}$and $\mathit{渭}\mathbf{=}\left(1,i\right)$for, $鈻�$.For the space ${\mathit{U}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$of upper triangular$\mathbf{2}\mathbf{脳}\mathbf{2}$matrices, use the basis

$\mathbf{渭}\mathbf{=}\mathbf{(}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{}\mathbf{)}$

Unless another basis is given. In each case, determine whether

Tis an isomorphism. If Tisn鈥檛 an isomorphism, find bases of the kernel and image of Tand thus determine the rank of T

13.role="math" localid="1659435121609" $\mathit{T}\left(M\right)\mathbf{=}\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right]\mathit{M}\mathbf{}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathbf{}{\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{}$ from to .

The linear transformation$\mathit{T}\left(f\right)\mathbf{=}\mathit{f}\mathbf{+}\mathit{f}\mathbf{\text{'}}\mathbf{\text{'}}$from${\mathit{C}}^{\mathbf{鈭�}}$to${\mathit{C}}^{\mathbf{鈭�}}$is an isomorphism.

In Exercises 5 through 40, find the matrix of the given linear transformationwith respect to the given basis. If no basis is specified, use standard basis:for,

forandfor,.For the spaceof upper triangularmatrices, use the basis

Unless another basis is given. In each case, determine whetheris an isomorphism. Ifisn鈥檛 an isomorphism, find bases of the kernel and image ofand thus determine the rank of.

13.fromto.

In Exercises 5 through 40, find the matrix of the given linear transformation with respect to the given basis. If no basis is specified, use standard basis: $\mathbf{渭}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{t}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{}\mathbf{for}\mathbf{}{\mathbf{P}}_{\mathbf{2}}$,

localid="1659445869146" $\mathit{渭}\mathbf{=}\mathbf{(}\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\mathbf{,}\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\mathbf{,}\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]\mathbf{,}\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\mathbf{)}$

for${\mathbf{鈻�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$and$\mathit{渭}\mathbf{=}\left(1,i\right)$for,$\mathbf{鈻�}$.For the space${\mathit{U}}^{\mathit{2}\mathit{脳}\mathit{2}}$of upper triangular$\mathbf{2}\mathbf{脳}\mathbf{2}$matrices, use the basis

$\mathit{渭}\mathbf{=}\mathbf{(}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{}\mathbf{)}$

Unless another basis is given. In each case, determine whether Tis an isomorphism. If Tisn鈥檛 an isomorphism, find bases of the kernel and image ofand thus determine the rank of T.

14. $\mathit{T}\left(M\right)\mathbf{=}\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right]\mathit{M}$ from ${\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$ to ${\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$with respect to the basis.

Find the transformation is linear and determine whether they are isomorphism.

All linear transformation from${\mathit{P}}_{\mathbf{3}}$to${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$are isomorphism.

Let Vbe the space of all infinite sequences of real numbers. See Example 5. Which of the subsets of Vgiven in Exercises 12 through 15 are subspaces of V? The sequences $\mathbf{(}{\mathbf{x}}_{\mathbf{0}}\mathbf{,}{\mathbf{x}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$that converge to zerorole="math" localid="1659412709897" $\mathbf{(}\mathbf{i}\mathbf{.}\mathbf{e}\mathbf{.}\mathbf{,}\underset{\mathbf{n}\mathbf{鈫�}\mathbf{鈭�}}{\mathbf{lim}\mathbf{}}{\mathbf{x}}_{\mathbf{n}}\mathbf{=}\mathbf{0}\mathbf{)}$