Exam 1: Linear Equations in Linear Algebra

Solve the problem. -Let $\mathrm { A } = \left[ \begin{array} { r r r } 2 & 3 & 1 \\ 8 & - 7 & 5 \end{array} \right]$ and $\mathbf { u } = \left[ \begin{array} { l } 3 \\ 7 \\ 2 \end{array} \right]$ . Define a transformation $\mathrm { T } : \mathcal { R } ^ { 3 } \rightarrow > \mathcal { R } ^ { 2 }$ by $\mathrm { T } ( \mathbf { x } ) = \mathrm { Ax }$ . Find $\mathrm { T } ( \mathbf { u } )$ , the image of $\mathbf { u }$ under the transformation $\mathrm { T }$ .

Determine whether the system is consistent. - ++=7 -+2=7 5++=11

Compute the product or state that it is undefined. - $\left[ \begin{array} { l l l } - 7 & 2 & 7\end{array} \right] \left[ \begin{array} { r } 3 \\0 \\- 3\end{array} \right]$

Solve the system of equations. - +3+2 =11 4+9 =-12 =-4

Display the indicated vector(s) on an xy-graph. - $\text { Let } \mathbf { u } = \left[ \begin{array} { r } - 3 \\5\end{array} \right] \text { and } \mathbf { v } = \left[ \begin{array} { r } - 2 \\2\end{array} \right] \text {. Display the vectors } \mathbf { u } , \mathbf { v } , \text { and } \mathbf { u } + \mathbf { v } \text { on the same axes. }$

Solve the problem. -The network in the figure shows the traffic flow (in vehicles per hour)over several one-way streets in the downtown area of a certain city during a typical lunch time. Determine the general flow Pattern for the network. In other words, find the general solution of the system of equations that describes the flow. In your General solution let x4 be free.

Determine whether the system is consistent. - +3+2=11 4+9=-12 +7+11=-11

Determine whether the system is consistent. - 5+2+=-11 2-3-=17 7-=12

Solve the problem. -Describe all solutions of Ax = b, where $A = \left[ \begin{array} { r r r } 2 & - 5 & 3 \\ - 2 & 6 & - 5 \\ - 4 & 7 & 0 \end{array} \right]$ and $\mathbf { b } = \left[ \begin{array} { r } - 3 \\ 4 \\ 3 \end{array} \right]$ Describe the general solution in parametric vector form.

Find the indicated vector. -Let $\mathbf { u } = \left[ \begin{array} { r } - 9 \\ 7 \end{array} \right] , \mathbf { v } = \left[ \begin{array} { r } - 3 \\ 4 \end{array} \right]$ . Find $\mathbf { u } + \mathbf { v }$ .

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise state that there is no solution. - $\left[ \begin{array} { r r r r } 1 & 0 & 6 & 3 \\0 & 1 & - 2 & - 3 \\0 & 0 & 0 & 0\end{array} \right]$

Determine whether the matrix is in echelon form, reduced echelon form, or neither. - $\left[ \begin{array} { r r r r } 1 & 0 & 0 & - 7 \\2 & 1 & 0 & - 2 \\0 & 5 & 1 & 2\end{array} \right]$

Determine whether the matrix is in echelon form, reduced echelon form, or neither. - $\left[ \begin{array} { r r r r } 1 & 2 & 5 & - 7 \\0 & 1 & - 4 & 9 \\0 & 0 & 1 & 2\end{array} \right]$

Determine whether the matrix is in echelon form, reduced echelon form, or neither. - $\left[ \begin{array} { r r r r } 1 & 4 & 5 & - 7 \\6 & 1 & - 4 & 8 \\0 & 5 & 1 & 6\end{array} \right]$

Find the indicated vector. -Let $\mathbf { u } = \left[ \begin{array} { r } 7 \\ - 1 \end{array} \right]$ . Find $- 6 \mathbf { u }$ .

Solve the problem. -For what values of h are the given vectors linearly independent? $\left[ \begin{array} { c } - 5 \\- 6 \\5\end{array} \right] , \left[ \begin{array} { c } 20 \\24 \\h\end{array} \right]$

Describe geometrically the effect of the transformation T. -Let $A = \left[ \begin{array} { l l l } 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ Define a transformation $\mathrm { T }$ by $\mathrm { T } ( \mathbf { x } ) = \mathrm { Ax }$ .

Solve the system of equations. - ++=7 -+2=7 5++=11

Solve the problem. -A company manufactures two products. For $1.00 worth of product A, the company spends $0.40 on materials, $0.25 on labor, and $0.10 on overhead. For $1.00 worth of product B, the company spends $0.50 on materials, $0.20 on labor, and $0.10 on overhead. Let $\mathbf { a } = \left[ \begin{array} { l } 0.40 \\0.25 \\0.10\end{array} \right] \text { and } \mathbf { b } = \left[ \begin{array} { l } 0.50 \\0.20 \\0.10\end{array} \right]$ Then a and b represent the ʺcosts per dollar of incomeʺ for the two products. Evaluate 100a + 400b and give an economic interpretation of the result.

Determine whether the system is consistent. - 4-+3 =12 2+9 =-5 +4+6 =-32