91Ó°ÊÓ

Are the following linear vector functions? Prove your conclusions using (7.2).

4.$\mathit{F}\mathbf{\left(}\mathit{r}\mathbf{\right)}\mathbf{=}\mathit{r}\mathbf{+}\mathit{A}$,whereAis a given vector.

Generalize Problem 6 to three dimensions; to n dimensions.

Prove the following by appropriate manipulations using Facts 1 to 4; do not just evaluate the determinants.

$\left|\begin{array}{ccc}1& a& bc\\ 1& b& ac\\ 1& c& ab\end{array}\right|\mathbf{=}\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|\mathbf{=}\left(c-a\right)\left(b-a\right)\left(c-b\right)\left|\begin{array}{ccc}1& a& a^2\\ 0& 1& b+a\\ 0& 0& 1\end{array}\right|\mathbf{=}\left(c-a\right)\left(b-a\right)\left(c-b\right)$

For each of the following sets, either verify (as in Example 1) that it is a vector space, or show which requirements are not satisfied. If it is a vector space, find a basis and the dimension of the space.

Polynomials of degree$\mathbf{≤}\mathbf{7}$with all the even coefficients equal to each other and all the odd coefficients equal to each other.

A line through the midpoint of one side of a triangle and parallel to a second side bisects the third side. Hint: Call parallel vectors $\stackrel{\mathbf{→}}{\mathbf{A}}$and $\mathit{c}\stackrel{\mathbf{→}}{\mathbf{A}}$.

Consider the set of numbers $\mathbf{1}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{7}\mathbf{}$with multiplication (mod 8) as the law of combination. Write the multiplication table to show that this is a group. [To multiply two numbers (mod 8); you multiply them and then take the remainder after dividing by 8. For examplerole="math" localid="1659338238237" $\mathbf{5}\mathbf{×}\mathbf{7}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{≡}\mathbf{3}\mathbf{(}\mathit{m}\mathit{o}\mathit{d}\mathbf{}\mathbf{8}\mathbf{)}$.) Is this group isomorphic to the cyclic group of order 4 or to the 4’s group?

Show that, in n-dimensional space, any $\mathbf{(}\mathit{n}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{)}$ vectors are linearly dependent. Hint: See Section 8.

Find the matrix product

$\left(23\right)\left(\begin{array}{cc}-1& 4\\ 2& -1\end{array}\right)\left(\begin{array}{c}-1\\ 2\end{array}\right)$

By evaluating this in two ways, verify the associative law for matrix multiplication, that is, A(BC)=(AB)C, which justifies our writing justABC.

Find the equations of the following conics and quadric surfaces relative to principal axes

${\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{3}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{3}{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{xy}\mathbf{+}\mathbf{4}\mathbf{xz}\mathbf{=}\mathbf{60}$

As in Problem 6,write$\mathbf{V}\mathbf{=}\mathbf{4}\mathbf{i}\mathbf{-}\mathbf{5}\mathbf{j}$ in terms of the basis vectors$\mathbf{i}\mathbf{-}\mathbf{4}\mathbf{j}$and$\mathbf{5}\mathbf{i}\mathbf{+}\mathbf{2}\mathbf{j}$.