91影视

Determine whether the statements that follow are true or false, and justify your answer.

15: The system$\mathbf{A}\stackrel{鈫�}{\mathbf{x}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{1}\end{array}\mathbf{\right]}$inconsistent for all $\mathbf{4}\mathbf{脳}\mathbf{3}$matrices A.

Compute the products Ax in Exercises 13 through 15 using

paper and pencil. In each case, compute the product

two ways: in terms of the columns of A and in terms of the rows of A.

15.

$\left[1234\right]\left[\begin{array}{c}5\\ 6\\ 7\\ 8\end{array}\right]$

Compute the products Ax in Exercises 16 through 19

using paper and pencil (if the products are defined).

16.

$\left[\begin{array}{cc}0& 1\\ 3& 2\end{array}\right]\left[\begin{array}{c}2\\ -3\end{array}\right]$

Let $\mathbf{T}\left(\stackrel{鈫�}{x}\right)\mathbf{=}{\mathbf{ref}}_{\mathbf{L}}\left(\stackrel{鈫�}{x}\right)$be the reflection about the line L in${\mathrm{鈩�}}^2$shown in the accompanying figure.

a. Draw sketches to illustrate that T is linear.

b. Find the matrix of T in terms of$\mathit{胃}$.

Solve the linear system of equations. You may use technology.

$\left|\begin{array}{c}3x\_1+6x\_2+9x\_3+5x\_4+25x\_5\&=53\\ 7x\_1+14x\_2+21x\_3+9x\_4+53x\_5\&=105\\ -4x\_1-8x\_2-12x\_3+5x\_4-10x\_5\&=11\end{array}\right|$

Determine whether the statements that follow are true or false, and justify your answer.

16: There exists a $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix such that

$\mathit{A}\left[\begin{array}{c}1\\ 1\end{array}\right]\mathbf{=}\left[\begin{array}{c}1\\ 2\end{array}\right]$and$\mathit{A}\left[\begin{array}{c}2\\ 2\end{array}\right]\mathbf{=}\left[\begin{array}{c}2\\ 1\end{array}\right]$

Question: Determine whether the statements that follow are true or false, and justify your answer.

17: Rank$\left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right|\mathbf{=}\mathbf{2}$.

Determine whether the statements that follow are true or false, and justify your answer.

17: Rank$\left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right|\mathbf{=}\mathbf{2}$

(For some background on the cross product in ${\mathbf{鈩�}}^{\mathbf{n}}$, see Exercise 6.2.44.) Consider three linearly independent vectors $\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{7}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}$ in ${\mathbf{鈩�}}^{\mathbf{4}}$.
a. What is the relationship between $\mathbf{V}\left(\stackrel{鈫�}{v_1},\stackrel{鈫�}{v_2},\stackrel{鈫�}{v_3}\right)$and $\mathbf{V}\mathbf{(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{)}$? See Definition 6.3.5. Exercise is helpful.
b. Express $\mathbf{V}\mathbf{(}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{,}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{1}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{2}}}\mathbf{脳}\stackrel{\mathbf{鈫�}}{{\mathbf{v}}_{\mathbf{3}}}\mathbf{)}$in terms ofrole="math" localid="1660118250452" $\left|{\stackrel{鈫�}{v}}_1脳{\stackrel{鈫�}{v}}_2脳{\stackrel{鈫�}{v}}_3\right|$.
c. Use parts (a) and (b) to express $\mathit{V}\left({\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3\right)$in terms of $||{\stackrel{鈫�}{v}}_1脳{\stackrel{鈫�}{v}}_2脳{\stackrel{鈫�}{v}}_3||$. Is your result still true when the ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathit{i}}$ are linearly dependent?
(Note the analogy to the fact that for two vectors ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$ and ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$ in role="math" localid="1660118435758" ${\mathbf{鈩�}}^{\mathbf{3}}||{\stackrel{鈫�}{v}}_1脳{\stackrel{鈫�}{v}}_2||$is the area of the parallelogram defined by${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$ and${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$.)

Compute the products Ax in Exercises 16 through 19

using paper and pencil (if the products are defined).

17.

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]\mathbf{.}\left[\begin{array}{c}7\\ 8\end{array}\right]$