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State true or false, if Vbea linear transformation from Vto V,then the intersection$\mathit{I}\mathit{m}\left(T\right)$and$\mathit{K}\mathit{e}\mathit{r}\left(T\right)$must be $\left\{0\right\}$$$" width="9" height="19" role="math">

Find the transformation is linear and determine whether they 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 square-summable sequences$\mathbf{(}{\mathit{x}}_{\mathbf{0}}\mathbf{,}{\mathit{x}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$ (i.e., those for which$\underset{\mathbf{i}\mathbf{=}\mathbf{0}}{\overset{\mathbf{鈭�}}{\mathbf{鈭�}}}{\mathit{x}}_{\mathbf{i}}^{\mathbf{2}}$converges).

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: for $\mathit{渭}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{t}\mathbf{,}\mathbf{t}\mathbf{)}$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 whether is an isomorphism. If isn鈥檛 an isomorphism, find bases of the kernel and image of and thus determine the rank of .

15. $\mathit{T}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{iy}\mathbf{)}\mathbf{=}\mathit{x}\mathbf{-}\mathit{i}\mathit{y}$ from C to C

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,$\mathit{T}\mathbf{\left(}\mathit{M}\mathbf{\right)}\mathbf{=}\mathit{M}\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]\mathbf{-}\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right]\mathit{M}$ from ${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}\mathbf{}{\mathbf{迟辞鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$to .

In Exercises 5 through 40, find the matrix of the given linear transformation $\mathit{T}$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 of Tand thus determine the rank of T.

16. $\mathit{T}\left(x-iy\right)\mathbf{=}\mathit{x}\mathbf{-}\mathit{i}\mathit{y}$ from Cto Cwith respect to the basis$\mathit{B}\mathbf{=}\left\{1+i,1-i\right\}$.

Find the basis of each of the space$\mathit{V}\mathbf{=}{\mathit{R}}^{\mathbf{3}\mathbf{脳}\mathbf{2}}$,and determine its dimension.

State true or false, the space of all upper triangular$\mathbf{4}\mathbf{脳}\mathbf{4}$matrices is isomorphic to the space of all lower triangular$\mathbf{4}\mathbf{脳}\mathbf{4}$matrices.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,T (x+iy) = x from C to C

Find the basis of each of the space $\mathit{V}\mathbf{=}{\mathit{R}}^{\mathbf{m}\mathbf{脳}\mathbf{n}}$, and determine its dimension.