91Ó°ÊÓ

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

48.If vector$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{w}\mathbf{}}$ is a linear combination of $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{u}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$ , then $\stackrel{\mathbf{â‡\P Ä}}{\mathbf{u}}\mathbf{+}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}\mathbf{+}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{w}\mathbf{}}$ must be a linear combination of$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{u}}$ and$\stackrel{\mathbf{â‡\P Ä}}{\mathbf{u}}\mathbf{+}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{v}}$ .

a. Find all solutions${\mathit{x}}_{\mathbf{1}}\mathbf{,}{\mathit{x}}_{\mathbf{2}}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{,}{\mathit{x}}_{\mathbf{4}}$ of the system .

${\mathit{x}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{3}}\mathbf{)}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{4}}\mathbf{)}$

b. In partrole="math" localid="1659677484607" $\mathbf{\left(}\mathit{a}\mathbf{\right)}$ , is there a solution with ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$and${\mathit{x}}_{\mathbf{4}}\mathbf{=}\mathbf{13}$ ?

Consider the accompanying table. For some linear systems$\mathbf{A}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$, you are given either the rank of the coefficient matrix$\mathbf{A}$ , or the rank of the augmented matrix $[A:\stackrel{→}{b}]$. In each case, state whether the system could have no solution, one solution, or infinitely many solutions. There may be more than one possibility for some systems. Justify your answers.

The eigenvalues of a symmetric matrix $\mathbf{A}$must be equal to the singular values of$\mathbf{A}$.

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

49.If the linear system$\mathit{A}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}$ has a unique solution and the linear system$\mathit{A}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{c}}$ is consistent, then the linear systemlocalid="1659962306596" $\mathit{A}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{+}\stackrel{\mathbf{â‡\P Ä}}{\mathbf{c}}$ must have a unique solution.

Find the rank of the matrices in 2 through 4.

4.$\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]$

in exercises 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions.

4.$$\begin{gathered} \left|2x+4y=2 \\ 3x+6y=3\right| \end{gathered}$$

Use the concept of a continuous dynamical system.Solve the differential equation$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{−}\mathit{k}\mathit{x}$. Solvethe system $\frac{d\stackrel{→}{x}}{dt}=A\stackrel{→}{x}$whenAis diagonalizable overR,and sketch the phase portrait for 2×2 matricesA.

Solve the initial value problems posed in Exercises 1through 5. Graph the solution.

4 .$\frac{dy}{dt}=0.8t$with$y\left(0\right)=-0.8$

For an arbitrary positive integer$\mathit{n}\mathbf{â‰\yen }\mathbf{3}$, find all solutions ${\mathit{x}}_{\mathbf{1}}\mathbf{,}{\mathit{x}}_{\mathbf{2}}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{,}\mathbf{.}\mathbf{.....}\mathbf{,}{\mathit{x}}_{\mathbf{n}}$of the simultaneous equations ${\mathit{x}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{3}}\mathbf{)}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{4}}\mathbf{)}\mathbf{,}\mathbf{.}\mathbf{....}\mathbf{,}{\mathit{x}}_{\mathbf{n}\mathbf{−}\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{n}\mathbf{−}\mathbf{2}}\mathbf{+}{\mathbf{x}}_{\mathbf{n}}\mathbf{)}$. Note that we are asked to solve the simultaneous equations ${\mathit{x}}_{\mathbf{k}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{(}{\mathbf{x}}_{\mathbf{k}\mathbf{−}\mathbf{1}}\mathbf{+}{\mathbf{x}}_{\mathbf{k}\mathbf{+}\mathbf{1}}\mathbf{)}$, for$\mathit{k}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{.}\mathbf{....}\mathbf{,}\mathit{n}\mathbf{−}\mathbf{1}$ .

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

50.A matrix is called a 0-1 matrix if all of its entries are ones and zeros. True or false: The majority of the 9-1 matrices of size 3X3 have rank 3.