91影视

T denotes the space of infinity sequence of real numbers,$\mathit{T}\left(f\left(t\right)\right)\mathbf{=}\left(t-1\right)\mathit{f}\left(t\right)$from P to P.

Question: If T is a linear transformation from ${\mathit{P}}_{\mathbf{6}}\mathbf{}\mathbf{to}\mathbf{}{\mathbf{P}}_{\mathbf{6}}$that transform${\mathbf{t}}^{\mathbf{k}}$ into a polynomial of degree$\mathit{k}\mathbf{(}\mathit{f}\mathit{o}\mathit{r}\mathbf{}\mathbf{}\mathit{k}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathbf{6}\mathbf{}\mathbf{)}$ (for ) then T must be an isomorphism鈥�.

a.Find the change of basis matrix S from the basis B considered in Exercise 28 to the standard basis$U=(1,t,t^2)$of$P_2$considered in Exercise 27.

b.Verify the formula SB=AS for the matrices B and A you found in Exercises 28 and 27,respectively.

c.Find the change of basis matrix from U to B

Question: There exist invertible $\mathbf{2}\mathbf{脳}\mathbf{2}$matrices P and Q such that the transformation$\mathbf{T}\left(f\right)\mathbf{=}\mathbf{PM}\mathbf{-}\mathbf{QM}$is an isomorphism.

Tdenotes the space of infinity sequence of real numbers,$\mathit{T}\left(f\left(t\right)\right)\mathbf{=}{\mathbf{鈭�}}_{\mathbf{0}}^{\mathbf{t}}\mathit{f}\left(x\right)\mathit{d}\mathit{x}$fromP to P.

A function$\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$from R to R is called even$\mathit{f}\mathbf{(}\mathbf{-}\mathbf{t}\mathbf{)}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$ iffor all t in R and odd if$\mathit{f}\mathbf{(}\mathbf{-}\mathbf{t}\mathbf{)}\mathbf{=}\mathbf{-}\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$for all t in R.Are the even functions a subspace of$\mathit{F}\mathbf{(}\mathbf{R}\mathbf{,}\mathbf{R}\mathbf{)}$,the space of all functions from R to R?What about the odd functions?Justify your answer carefully.

In Exercises 48 through 53, let V be the space spanned by the two functions$\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$and $\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$. In each exercise, find the matrix of the given transformation T with respect to the basis $\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{,}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$, and determine whether T is an isomorphism. $\mathit{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\text{'}}$

Question: There exists a linear transformation from ${\mathbf{P}}_{\mathbf{6}}$to Cwhose kernel is isomorphic to${\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$.

Tdenotes the space of infinity sequence of real numbers,$\mathit{T}\left(f\left(t\right)\right)\mathbf{=}{\mathit{f}}^{\mathbf{\text{'}}}\left(t\right)$fromP to P.

Find a basis of each of the following linear spaces and thus determine their dimensions.See Exercise 47.

$\begin{array}{l}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{\{}\mathbf{f}\mathbf{\text{鈥夆赌�}}\mathbf{i}\mathbf{n}\mathbf{\text{鈥夆赌夆��}}{\mathbf{P}}_{\mathbf{4}}\mathbf{:}\mathbf{\text{鈥�}}\mathbf{f}\mathbf{\text{鈥�}}\mathbf{i}\mathbf{s}\mathbf{\text{鈥�}}\mathbf{e}\mathbf{v}\mathbf{e}\mathbf{n}\mathbf{\}}\\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{\{}\mathbf{f}\mathbf{\text{鈥夆赌�}}\mathbf{i}\mathbf{n}\mathbf{\text{鈥夆赌夆��}}{\mathbf{P}}_{\mathbf{4}}\mathbf{:}\mathbf{\text{鈥�}}\mathbf{f}\mathbf{\text{鈥�}}\mathbf{i}\mathbf{s}\mathbf{\text{鈥�}}\mathbf{o}\mathbf{d}\mathbf{d}\mathbf{\}}\end{array}$