91影视

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $\mathit{T}\left(f\left(t\right)\right)\mathbf{=}\mathit{F}\mathbf{\text{'}}\mathbf{\text{'}}\left(t\right)\mathit{f}\left(t\right)$ from${\mathit{P}}_{\mathbf{2}}$to ${\mathit{P}}_{\mathbf{2}}$.

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

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

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

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

Unless another basis is given. In each case, determine whetherTis an isomorphism. IfTisn鈥檛 an isomorphism, find bases of the kernel and image ofTand thus determine the rank ofT.

24.$\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{3}\mathbf{\right)}$from${\mathit{P}}_{\mathbf{2}}$ to${\mathit{P}}_{\mathbf{2}}$ with respect to the basis$\mathit{尾}\mathbf{=}\left\{1,t-3,{\left(t-3\right)}^2\right\}$.

Find the basis of all$\mathbf{3}\mathbf{脳}\mathbf{3}$ upper triangular matrix, and determine its dimension.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $\mathit{T}\left(f\left(t\right)\right)\mathbf{=}\mathit{f}\mathit{"}\left(t\right)\mathit{+}\mathit{4}\mathit{f}\mathit{\text{'}}\left(t\right)$from ${\mathit{P}}_{\mathit{2}}\mathbf{to}{\mathit{P}}_{\mathit{2}}$.

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

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

for localid="1659755272536" ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$andfor,$\mathit{渭}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{i}\mathbf{)}\mathbf{}\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathbf{,}\mathbf{鈩傗刚鈩�}$ .For the space ${\mathit{U}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$of upper triangular $\mathbf{2}\mathbf{脳}\mathbf{2}$matrices, use the basis

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

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.

22.$\mathit{T}\left(t\right)\mathbf{=}\mathit{f}\mathbf{(}\mathbf{-}\mathit{t}\mathbf{)}\mathbf{}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathbf{}{\mathit{P}}_{\mathbf{2}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathit{P}}_{\mathbf{2}}$ .

Find the set of all polynomial$\mathit{f}\left(t\right)$ in${\mathit{P}}_{\mathbf{2}}$ such that$\mathit{f}\left(1\right)\mathbf{=}\mathbf{0}$, and determine its dimension.

State true or false, there exist $\mathbf{2}\mathbf{脳}\mathbf{2}$ matrix$\mathit{A}$ such that the space$\mathit{V}$ of all matrices commuting with$\mathit{A}$ is two-dimensional.

State true or false, there exist a basis of ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$that consist of four invertible matrices.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $\mathit{T}\left(f\left(t\right)\right)\mathbf{=}\mathit{f}\left(-t\right)$from ${\mathit{P}}_{\mathbf{2}}$to ${\mathit{P}}_{\mathbf{2}}$that is, $\mathit{T}\left(a+bt+c{t}^4\right)\mathbf{=}\mathit{a}\mathbf{-}\mathit{b}\mathit{t}\mathbf{+}\mathit{c}{\mathit{t}}^{\mathbf{t}}$

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

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

for ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$and $\mathit{渭}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{i}\mathbf{)}\mathbf{}\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathbf{,}\mathbf{鈩�}$.For the space${\mathit{U}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$of upper triangular 2x2matrices, use the basis

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

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.

26. role="math" localid="1659760276807" $\mathit{T}\mathbf{\left(}\mathit{f}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{2}\mathit{t}\mathbf{\right)}\mathbf{}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathbf{}{\mathit{P}}_{\mathbf{2}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathit{P}}_{\mathbf{2}}$.