91影视

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

Find a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace. Which of the subsets of given in Exercises 1through 5are subspaces of${\mathit{P}}_{\mathbf{2}}$ (see Example 16)? Find a basis for those that are subspaces,$\left\{p\left(t\right):p\left(0\right)=2\right\}$.

TRUE OR FALSE?

There exist$\mathbf{2}\mathbf{脳}\mathbf{2}$matrix Asuch that the space Vof all matrices commuting with Ais one-dimensional.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,$\mathit{T}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}$from$\mathbf{鈩�}$ to $\mathbf{鈩�}$.

Find the basis of all matrices $\mathit{A}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\mathbf{}\mathbf{in}\mathbf{}{\mathbf{R}}^{\mathbf{2}\mathbf{x}\mathbf{2}}$in such that a+d=0,and determine its dimension.

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: $\mathit{渭}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathit{t}\mathbf{,}\mathit{t}\mathbf{)}$for ${\mathbf{P}}_{\mathbf{2}}$,

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

for ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$and$\mathit{渭}\mathbf{=}\left(1,i\right)$for, $\mathbf{鈩�}$.For the space${\mathbf{U}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$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 whetherTis an isomorphism. IfTisn鈥檛 an isomorphism, find bases of the kernel and image ofTand thus determine the rank ofT.

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

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,$\mathit{T}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{)}\mathbf{=}\mathit{y}\mathbf{+}\mathit{i}\mathit{x}$
from to .

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

21.$\mathit{T}\left(f\right)\mathbf{=}\mathit{f}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathit{f}$ from${\mathit{P}}_{\mathbf{2}}$ to${\mathit{P}}_{\mathbf{2}}$ with respect to the basis$\mathit{B}\mathbf{=}\left\{1,t,t^2\right\}$.

Find the basis of all 2X2diagonal matrix,and determine its dimension.