91影视

TRUE OR FALSE?

38. There exists a subspace of${\mathbf{鈩�}}^{\mathbf{3}\mathbf{脳}\mathbf{4}}$that is isomorphic to${\mathit{P}}_{\mathbf{9}}$.

If B is a diagonal $\mathbf{4}\mathbf{脳}\mathbf{4}$matrix, what are the possible dimensions of the space V of all data-custom-editor="chemistry" $\mathbf{4}\mathbf{脳}\mathbf{4}$matrices A that commute with B?

Find the transformation is linear and determine whether the transformation is an isomorphism.

What is the dimensions of the space of all upper triangular$\mathbf{n}\mathbf{脳}\mathbf{n}$ matrices?

Find the transformation is linear and determine whether the transformation is an isomorphism.

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 the standard basis: $\mathit{u}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{t}\mathbf{,}{\mathbf{t}}^{\mathbf{2}}\mathbf{)}$for ${\mathit{P}}_{\mathbf{2}}$,

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

for${\mathit{R}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$, and$\mathbf{}\mathit{u}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{i}\mathbf{)}$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

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

unless another basis is given. In each case, determine whether T is an isomorphism. If T isn鈥檛 an isomorphism, find bases of the kernel and image of T, and thus determine the rank of T.

39.$\mathit{T}\mathbf{\left(}\mathbf{M}\mathbf{\right)}\mathbf{=}\mathbf{\left[}\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}\end{array}\mathbf{\right]}\mathit{M}\mathbf{-}\mathit{M}\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}\end{array}\mathbf{\right]}$from${\mathit{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$to ${\mathit{鈩�}}^{\mathbf{2}\mathbf{脳}\mathbf{2}}$

Do the polynomials $\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{2}\mathit{t}\mathbf{+}\mathbf{9}{\mathit{t}}^{\mathbf{2}}\mathbf{+}{\mathit{t}}^{\mathbf{3}}\mathbf{,}$$\mathit{g}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{7}\mathit{t}\mathbf{+}\mathbf{7}{\mathit{t}}^{\mathbf{3}}\mathbf{,}$$\mathit{h}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{8}\mathit{t}\mathbf{+}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathbf{5}{\mathit{t}}^{\mathbf{3}}\mathbf{,}$$\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{k}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{8}\mathit{t}\mathbf{+}\mathbf{4}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathbf{8}{\mathit{t}}^{\mathbf{3}}$

form a basis of${\mathit{P}}_{\mathbf{3}}$?

TRUE OR FALSE?

3. The lower triangular $\mathbf{2}\mathbf{脳}\mathbf{2}$matrices from a subspace of the space of all$\mathbf{2}\mathbf{脳}\mathbf{2}$matrices.

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 ${\mathit{P}}_{\mathbf{2}}$given in Exercises 1throughare subspaces of ${\mathit{P}}_{\mathbf{2}}$ (see Example)? Find a basis for those that are subspaces, $\left\{p\left(t\right):p^{\text{'}}\left(1\right)=p\left(2\right)\right\}\mathbf{(}{\mathit{p}}^{\mathbf{\text{'}}}\mathit{i}\mathit{s}\mathit{t}\mathit{h}\mathit{e}\mathit{d}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{.}\mathbf{)}$