91影视

True or False.

The linear transformation $\mathbf{T}\left(M\right)\mathbf{=}\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right]\mathbf{M}\mathbf{}\mathbf{from}\mathbf{}{\mathbf{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{}\mathbf{to}\mathbf{}{\mathbf{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{}\mathbf{}\mathbf{have}\mathbf{}\mathbf{rank}\mathbf{}\mathbf{1}\mathbf{.}$

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,$\mathit{T}\left(f\left(t\right)\right)\mathbf{=}{\mathbf{鈭�}}_{\mathbf{-}\mathbf{2}}^{\mathbf{3}}\mathit{f}\left(t\right)\mathit{d}\mathit{t}$ from to${\mathit{P}}_{\mathbf{2}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}\mathbf{鈩�}$.

Find the basis of all nxn diagonal matrix, and determine its dimension.

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:$\mathit{渭}\mathbf{=}\left(1,t,t\right)$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${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$and$\mathit{渭}\mathbf{=}\left(1,i\right)$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 whether Tis an isomorphism. If Tisn鈥檛 an isomorphism, find bases of the kernel and image ofTand thus determine the rank ofT.

22.$\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{=}{\mathbf{f}}^{\mathbf{\text{'}\text{'}}}\mathbf{+}\mathbf{4}{\mathbf{f}}^{\mathbf{\text{'}}}$from${\mathit{P}}_{\mathbf{2}}$ to ${\mathit{P}}_{\mathbf{2}}$.

True or false, if the matrix of a linear transformation T(with respect to some basis) is $\left[\begin{array}{cc}3& 5\\ 0& 4\end{array}\right]$, then there must exist a nonzero element f in the domain of T such that T(f)=3f.

Find the basis of all 2x2 lower triangular matrix, and determine its dimension.

The kernel of liner transformation$\mathbf{T}\left(f\left(t\right)\right)\mathbf{=}\mathbf{f}\left(t^2\right)\mathbf{}\mathbf{from}\mathbf{}\mathbf{P}\mathbf{}\mathbf{to}\mathbf{}\mathbf{P}\mathbf{}\mathbf{is}\mathbf{}\left\{0\right\}$.

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${\mathit{P}}_{\mathbf{2}}$,

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

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.

23.$\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{=}\mathit{f}\left(3\right)$from${\mathit{P}}_{\mathbf{2}}$ to ${\mathit{P}}_{\mathbf{2}}$.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $\mathit{T}\mathbf{\left(}\mathbf{f}\left(t\right)\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{7}\mathbf{\right)}$ from${\mathit{P}}_{\mathbf{2}}$torole="math" localid="1659412169328" $\mathbf{鈩�}$.

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$\mathit{尾}\mathbf{=}\left\{1,t-3,{\left(t-3\right)}^2\right\}$.