Exam 1: Linear Equations in Linear Algebra

Solve the problem. -The columns of $\mathrm { I } _ { 3 } = \left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$ are $\mathbf { e } _ { 1 } = \left[ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right] , \mathbf { e } _ { 2 } = \left[ \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right] , \mathbf { e } _ { 3 } = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right]$ . Suppose that $\mathrm { T }$ is a linear transformation from $R ^ { 3 }$ into $\mathcal { R } ^ { 2 }$ such that $\mathrm { T } \left( \mathbf { e } _ { 1 } \right) = \left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] , \mathrm { T } \left( \mathbf { e } _ { 2 } \right) = \left[ \begin{array} { l } 5 \\ 0 \end{array} \right]$ , and $\mathrm { T } \left( \mathbf { e } _ { 3 } \right) = \left[ \begin{array} { r } - 5 \\ 1 \end{array} \right]$ Find a formula for the image of an arbitrary $x = \left[ \begin{array} { l } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \end{array} \right]$ in $R ^ { 3 }$ .

Solve the problem. -Let $\mathbf { a } _ { 1 } = \left[ \begin{array} { r } 3 \\ 4 \\ - 4 \end{array} \right] , \mathbf { a } _ { 2 } = \left[ \begin{array} { r } - 4 \\ 1 \\ 1 \end{array} \right]$ , and $\mathbf { b } = \left[ \begin{array} { r } 2 \\ - 10 \\ 6 \end{array} \right]$ Determine whether $\mathbf { b }$ can be written as a linear combination of $\mathbf { a } _ { \mathbf { 1 } }$ and $\mathbf { a } _ { \mathbf { 2 } }$ . In other words, determine whether weights $x _ { 1 }$ and $x _ { 2 }$ exist, such that $x _ { 1 } a _ { 1 } + x _ { 2 } a _ { 2 } = b$ . Determine the weights $x _ { 1 }$ and $x _ { 2 }$ if possible.

Determine whether the matrix is in echelon form, reduced echelon form, or neither. - $\left[ \begin{array} { r r r r } 1 & - 5 & 3 & 4 \\0 & 0 & - 5 & - 4 \\0 & 0 & 0 & - 2 \\0 & 0 & 0 & 0\end{array} \right]$

Solve the problem. - $2 x _ { 1 } - 16 x _ { 2 } + 10 x _ { 3 } = 0$

Solve the problem. -Let $T : R ^ { 2 } \rightarrow \mathscr { R } ^ { 2 }$ be a linear transformation that maps $\mathbf { u } = \left[ \begin{array} { r } - 6 \\ 4 \end{array} \right]$ into $\left[ \begin{array} { r } - 22 \\ 12 \end{array} \right]$ and maps $\mathbf { v } = \left[ \begin{array} { r } 2 \\ - 5 \end{array} \right]$ into $\left[ \begin{array} { c } 11 \\ - 4 \end{array} \right]$ . Use the fact that $T$ is linear to find the image of $3 \mathbf { u } + \mathbf { v }$ .

Write the system as a vector equation or matrix equation as indicated. -Write the following system as a matrix equation involving the product of a matrix and a vector on the left side and a vector on the right side. 2+-6 =-6 6-4 =2

Find the standard matrix of the linear transformation T. -T: $R ^ { 2 } \rightarrow R ^ { 2 }$ first performs a vertical shear that maps $\mathbf { e } _ { 1 }$ into $\mathbf { e } _ { 1 } + 3 \mathbf { e } _ { 2 }$ , but leaves the vector $\mathbf { e } _ { 2 }$ unchanged, then reflects the result through the horizontal $x _ { 1 }$ -axis.

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. - $\left[ \begin{array} { r r r r } 1 & 2 & - 3 & - 6 \\0 & 1 & 4 & 7 \\0 & 0 & 0 & 2\end{array} \right]$

Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated. -Find the reduced echelon form of the given matrix. $\left[ \begin{array} { c r r r r } 1 & 4 & - 5 & 1 & 2 \\2 & 5 & - 4 & - 1 & 4 \\- 3 & - 9 & 9 & 2 & 10\end{array} \right]$

Determine whether the matrix is in echelon form, reduced echelon form, or neither. - $\left[ \begin{array} { r r r r } 1 & 0 & - 3 & - 5 \\0 & 1 & - 3 & 4 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{array} \right]$

Determine whether the matrix is in echelon form, reduced echelon form, or neither. - $\left[ \begin{array} { r r r r } 1 & 4 & 5 & - 7 \\0 & 1 & - 4 & - 6 \\0 & 2 & 1 & 6\end{array} \right]$

Solve the problem. -Suppose an economy consists of three sectors: Energy (E), Manufacturing (M), and Agriculture (A) Sector E sells $70 \%$ of its output to $\mathrm { M }$ and $30 \%$ to $\mathrm { A }$ . Sector M sells $30 \%$ of its output to E, $50 \%$ to A, and retains the rest. Sector A sells $15 \%$ of its output to E, $30 \%$ to $M$ , and retains the rest. Denote the prices (dollar values) of the total annual outputs of the Energy, Manufacturing, and Agriculture sectors by $\mathrm { p } _ { \mathrm { e } } , \mathrm { p } _ { \mathrm { m } }$ , and $\mathrm { p } _ { \mathrm { a } }$ , respectively. If possible, find equilibrium prices that make each sector's income match its expenditures. Find the general solution as a vector, with $\mathrm { p } _ { \mathrm { a } }$ free.

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

Solve the problem. -The table shows the amount (in g)of protein, carbohydrate, and fat supplied by one unit (100 g)of three different foods. Let $\mathrm { T }$ be the linear transformation whose standard matrix is $A = \left[ \begin{array} { r r r } 1 & - 2 & 3 \\- 1 & 3 & - 4 \\- 2 & - 9 & 5\end{array} \right]$ Determine whether the linear transformation $\mathrm { T }$ is one-to-one and whether it maps $R ^ { 3 }$ onto $R ^ { 3 }$ .

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

Determine whether the matrix is in echelon form, reduced echelon form, or neither. - $\left[ \begin{array} { r r r r } 1 & 6 & 2 & - 7 \\0 & 1 & - 4 & - 6 \\0 & 0 & 0 & 0\end{array} \right]$

Solve the system of equations. - 7+7+=1 +8+8=8 9++9=9

Solve the problem. -Let $A = \left[ \begin{array} { r r r } 1 & - 3 & 2 \\ - 3 & 4 & - 1 \\ 2 & - 5 & 3 \end{array} \right]$ and $\mathbf { b } = \left[ \begin{array} { r } 2 \\ 4 \\ - 4 \end{array} \right]$ . Define a transformation $\mathrm { T } : \mathfrak { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }$ by $\mathrm { T } ( \mathrm { x } ) = \mathrm { Ax }$ . If possible, find a vector $\mathbf { x }$ whose image under $\mathrm { T }$ is $\mathbf { b }$ . Otherwise, state that $\mathbf { b }$ is not in the range of the transformation $\mathrm { T }$ .

Solve the problem. -Find the general solution of the homogeneous system below. Give your answer as a vector. +2-3=0 4+7-9=0 --4+9=0

Find the indicated vector. -Let $\mathbf { u } = \left[ \begin{array} { r } - 1 \\ 1 \end{array} \right] , \mathbf { v } = \left[ \begin{array} { r } - 4 \\ 3 \end{array} \right]$ . Find $\mathbf { u } - \mathbf { v }$ .