91影视

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 $\mathit{渭}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{i}\mathbf{)}$for, $\mathbf{鈩傗刚}$ .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.

29. $\mathit{T}\left(f\right)\mathbf{=}{\mathbf{鈭�}}_{\mathbf{0}}^{\mathbf{2}}\mathit{f}\left(t\right)\mathit{d}\mathit{t}\mathbf{}\mathit{f}\mathit{r}\mathit{o}\mathit{m}\mathbf{}{\mathit{P}}_{\mathbf{2}}\mathbf{}\mathit{t}\mathit{o}\mathbf{}{\mathit{P}}_{\mathbf{2}}$.

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

State true or false, if a, b andare distinct real numbers then the polynomials a basis of linear space V then$\left(x-b\right)\left(x-c\right)$,$\left(x-a\right)\left(x-c\right)$ and$\left(x-a\right)\left(x-b\right)$ must be a linear independent.

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

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

TRUE OR FALSE?

The function$\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{=}\mathbf{3}\mathbf{f}\mathbf{-}\mathbf{4}\mathbf{f}\mathbf{\text{'}}$from ${\mathbf{C}}^{\mathbf{伪}}$to${\mathbf{C}}^{\mathbf{伪}}$is a linear transformation.

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${\mathit{P}}_{\mathbf{2}}$ given in Exercises 1through 5are subspaces ofrole="math" localid="1659355918761" ${\mathit{P}}_{\mathbf{2}}$ (see Example)? Find a basis for those that are subspaces,$\left\{p\left(t\right):p\left(2\right)=0\right\}$.

Find the basis of all$\mathbf{2}\mathbf{脳}\mathbf{2}$matrix A such that $\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{2}\\ \mathbf{3}& \mathbf{6}\end{array}\mathbf{\right]}A\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}$, and determine itsdimension.

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

State true or false, the linear transformation T (f) = f (4t - 3 ) from P toP is an isomorphism.