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Find the set of all polynomial $\mathit{f}\left(t\right)$in ${\mathit{P}}_{\mathbf{3}}$ such that $\mathit{f}\left(1\right)\mathbf{=}\mathbf{0}$ and${\mathbf{鈭�}}_{\mathbf{-}\mathbf{1}}^{\mathbf{1}}\mathit{f}\left(t\right)\mathit{d}\mathit{t}\mathbf{=}\mathbf{0}$,and determine its dimension.

In Exercises 5 through 40, find the matrix of the given linear transformation T with respect to the given basis. If no basis is specified, use standard basis: $\mathit{渭}\mathbf{=}\left(1t,t\right)$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 triangularrole="math" localid="1659506679158" $\mathbf{2}\mathbf{脳}\mathbf{2}$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.

27.$\mathit{T}\left(f\right)\mathbf{=}\mathit{f}\left(2t-1\right)$ 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, $\mathit{T}\left(f\left(t\right)\right)\mathbf{=}\mathit{f}\left(2t\right)$ from${\mathit{P}}_{\mathbf{2}}$to ${\mathit{P}}_{\mathbf{2}}$that is,role="math" localid="1659780998385" $\mathit{T}\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{bt}\mathbf{+}{\mathbf{ct}}^{\mathbf{2}}\mathbf{)}\mathbf{=}\mathit{a}\mathbf{-}\mathit{b}\mathit{t}\mathbf{+}\mathit{c}{\mathit{t}}^{\mathbf{t}}$.

Find the basis of all$\mathbf{2}\mathbf{脳}\mathbf{2}$ matrixA such thatA commute with $\mathit{B}\mathbf{=}\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$, and determine its dimension.

State true or false, if the image of a linear transformation T from P to P is all P, thenmust be an isomorphism.

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

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

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

for${\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}{\mathbf{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$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 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.

28. $\mathit{T}\left(t\right)\mathbf{=}\mathit{f}\left(2t-1\right)$ from ${\mathit{p}}_{\mathbf{2}}$ to ${\mathit{p}}_{\mathbf{2}}$ with respect to the basis $\mathit{B}\mathbf{=}\left\{1,t-1,{\left(t-1\right)}^2\right\}$.

State true or false, if ${\mathbf{f}}_{\mathbf{1}}$,${\mathbf{f}}_{\mathbf{2}}$ and${\mathbf{f}}_{\mathbf{3}}$is a basis of linear space V then${\mathbf{f}}_{\mathbf{1}}$,${\mathbf{f}}_{\mathbf{2}}\mathbf{+}{\mathbf{f}}_{\mathbf{3}}$ and${\mathbf{f}}_{\mathbf{1}}\mathbf{+}{\mathbf{f}}_{\mathbf{2}}\mathbf{+}{\mathbf{f}}_{\mathbf{3}}$ must be a basis of Vas well.

Find the basis of all $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix A such that A commute with $\mathit{B}\mathbf{=}\left[\begin{array}{cc}\mathit{1}& \mathit{1}\\ \mathit{0}& \mathit{1}\end{array}\right]$,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, $\mathit{T}\mathbf{\left(}\mathit{f}\mathbf{\right(}\mathit{t}\mathbf{\left)}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{2}\mathit{t}\mathbf{\right)}\mathbf{-}\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ from${\mathit{P}}_{\mathbf{2}}$ to ${\mathit{P}}_{\mathbf{2}}$.