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Let V be the space of all infinite sequences of real numbers. See Example 5. Which of the subsets of given in Exercises 12 through 15 are subspaces of V ? The arithmetic sequences [i.e., sequences of the form$\mathbf{(}\mathbf{a}\mathbf{,}\mathbf{a}\mathbf{+}\mathbf{k}\mathbf{,}\mathbf{a}\mathbf{+}\mathbf{2}\mathbf{k}\mathbf{,}\mathbf{a}\mathbf{+}\mathbf{3}\mathbf{k}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{)}$ , for some constants and K .

TRUE OR FALSE?

Find the basis of all 2x2 lower triangular matrix, and determine its dimension.

Find the basis of all$\mathbf{3}\mathbf{×}\mathbf{3}$ upper triangular matrix, and determine its dimension.

a.Find the change of basis matrix S from the basis B considered in Exercise 24 to the standard basis $\mathit{U}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{t}\mathbf{,}{\mathbf{t}}^{\mathbf{2}}\mathbf{)}$of ${\mathit{P}}_{\mathbf{2}}$considered in Exercise 23.

b.Verify the formula SB = AS for the matrices B and A you found in Exercises 24 and 23,respectively.

c.Find the change of basis matrix from $\mathfrak{U}\mathbf{}\mathit{t}\mathit{o}\mathbf{}\mathfrak{B}$.

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 ${\mathbf{P}}_{\mathbf{2}}$given in Exercises 1through 5are subspaces of${\mathbf{P}}_{\mathbf{2}}$ (see Example 16)? Find a basis for those that are subspaces,$\mathbf{\{}\mathbf{p}\left(t\right)\mathbf{:}\mathbf{p}\left(-t\right)\mathbf{=}\mathbf{-}\mathbf{p}\left(t\right)\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{all}\mathbf{}\mathbf{t}\mathbf{\}}$.