Exam 1: Linear Equations in Linear Algebra

Solve the problem. -Let $A = \left[ \begin{array} { r r r } 1 & - 3 & 2 \\ - 2 & 5 & - 1 \\ 3 & - 6 & - 3 \end{array} \right]$ and $\mathbf { b } = \left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right]$ Determine if the equation $\mathrm { Ax } = \mathrm { b }$ is consistent for all possible $\mathrm { b } _ { 1 } , \mathrm {~b} _ { 2 } , \mathrm {~b} _ { 3 }$ . If the equation is not consistent for all possible $b _ { 1 } , b _ { 2 } , b _ { 3 }$ , give a description of the set of all $\mathbf { b }$ for which the equation is consistent (i.e., a condition which must be satisfied by $b _ { 1 } , b _ { 2 } , b _ { 3 }$ ).

Write the system as a vector equation or matrix equation as indicated. -Write the following system as a vector equation involving a linear combination of vectors. 6-6-=5 6+3=-5

Solve the problem. - $\mathrm { T } \left( \mathrm { x } _ { 1 } , \mathrm { x } _ { 2 } , \mathrm { x } _ { 3 } \right) = \left( - 4 \mathrm { x } _ { 2 } - 4 \mathrm { x } _ { 3 } , - 2 \mathrm { x } _ { 1 } + 8 \mathrm { x } _ { 2 } + 4 \mathrm { x } _ { 3 } , - \mathrm { x } _ { 1 } - 2 \mathrm { x } _ { 3 } , 4 \mathrm { x } _ { 2 } + 4 \mathrm { x } _ { 3 } \right)$ Determine whether the linear transformation $\mathrm { T }$ is one-to-one and whether it maps $\mathfrak { R } ^ { 3 }$ onto $\mathfrak { R } ^ { 4 }$ .

Solve the system of equations. - -+=8 ++=6 +-=-12

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. - $\left[ \begin{array} { r r r } 1 & - 5 & - 1 \\0 & 0 & 7\end{array} \right]$

Solve the system of equations. - ++=6 ==-2 +3=11

Determine whether the linear transformation T is one-to-one and whether it maps as specified. - Betty would like to prepare a meal using some combination of these three foods. She would like the meal to contain $15 \mathrm {~g}$ of protein, $25 \mathrm {~g}$ of carbohydrate, and $3 \mathrm {~g}$ of fat. How many units of each food should she use so that the meal will contain the desired amounts of protein, carbohydrate, and fat? Round to 3 decimal places.

Find the indicated vector. -Let $\mathbf { u } = \left[ \begin{array} { l } 2 \\ 7 \end{array} \right] , \mathbf { v } = \left[ \begin{array} { l } - 8 \\ - 5 \end{array} \right]$ . Find $\mathbf { v } - \mathbf { u }$ .

Solve the system of equations. - -+3 =-8 2+ =0 +5+ =40

Solve the problem. -For what values of h are the given vectors linearly dependent? 66) $\left[ \begin{array} { r } - 1 \\4 \\6\end{array} \right] , \left[ \begin{array} { r } 5 \\2 \\- 3\end{array} \right] , \left[ \begin{array} { r } - 1 \\1 \\1\end{array} \right] , \left[ \begin{array} { r } 2 \\- 2 \\h\end{array} \right]$

Solve the system of equations. - 4-+3 =12 2+9 =-5 +4+6 =-32

Determine whether the linear transformation T is one-to-one and whether it maps as specified. -The population of a city in 2000 was 400,000 while the population of the suburbs of that city in 2000 was 900,000. Suppose that demographic studies show that each year about $5 \%$ of the city's population moves to the suburbs (and $95 \%$ stays in the city), while $4 \%$ of the suburban population moves to the city (and $96 \%$ remains in the suburbs). Compute the population of the city and of the suburbs in the year 2002. For simplicity, ignore other influences on the population such as births, deaths, and migration into and out of the city/suburban region.

Determine whether the system is consistent. - + +=6 -=-2 +3 =11

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. - $\left[ \begin{array} { r r r r r r } 1 & 5 & 8 & - 1 & 2 & 5 \\0 & 0 & 0 & - 4 & 3 & 4 \\0 & 0 & 0 & 0 & - 2 & 8\end{array} \right]$

Solve the problem. -Let $\mathrm { A } = \left[ \begin{array} { r r r } 1 & - 3 & 0 \\ - 4 & 0 & 2 \\ 4 & 1 & - 2 \end{array} \right]$ and $\mathbf { b } = \left[ \begin{array} { r } 9 \\ - 6 \\ 4 \end{array} \right]$ . Define a transformation $\mathrm { T }$ : $\mathfrak { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }$ by $\mathrm { T } ( \mathbf { x } ) = \mathrm { Ax }$ . If possible, find a vector $x$ whose image under $\mathrm { T }$ is $\mathbf { b }$ . Otherwise, state that $\mathbf { b }$ is not in the range of the transformation $\mathrm { T }$ .

Compute the product or state that it is undefined. - $\left[ \begin{array} { r r } - 1 & 3 \\- 8 & - 5 \\- 6 & - 8\end{array} \right] \left[ \begin{array} { r } 3 \\- 1\end{array} \right]$

Find the indicated vector. -Let $\mathbf { u } = \left[ \begin{array} { l } - 9 \\ - 2 \end{array} \right]$ . Find $- 3 \mathbf { u }$ .

Solve the problem. -Let $\mathbf { v } _ { \mathbf { 1 } } = \left[ \begin{array} { r } 1 \\ - 3 \\ 5 \end{array} \right] , \mathbf { v } _ { \mathbf { 2 } } = \left[ \begin{array} { r } - 3 \\ 8 \\ 3 \end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 2 \\ - 2 \\ - 6 \end{array} \right]$ Determine if the set $\left\{ \mathbf { v } _ { \mathbf { 1 } } , \mathbf { v } _ { \mathbf { 2 } } , \mathbf { v } _ { \mathbf { 3 } } \right\}$ is linearly independent.

Determine whether the system is consistent. - -+4 =15 -4+4-16 =4 +4+ =0