91Ó°ÊÓ

In Exercises \(1-10\) , assume that \(T\) is a linear transformation. Find the standard matrix of \(T\) . \(T : \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}, \quad T\left(\mathbf{e}_{1}\right)=(1,3), \quad T\left(\mathbf{e}_{2}\right)=(4,-7), \quad\) and \(T\left(\mathbf{e}_{3}\right)=(-5,4),\) where \(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\) are the columns of the \(3 \times 3\) identity matrix.

Balance the chemical equations in Exercises \(5-10\) using the vector equation approach discussed in this section. Alka-Seltzer contains sodium bicarbonate \(\left(\mathrm{NaHCO}_{3}\right)\) and citric acid \(\left(\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}\right) .\) When a tablet is dissolved in water, the following reaction produces sodium citrate, water, and carbon dioxide (gas): \(\mathrm{NaHCO}_{3}+\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7} \rightarrow \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}+\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2}\)

In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system. $$ \left[\begin{array}{rrrr}{1} & {-4} & {9} & {0} \\ {0} & {1} & {7} & {0} \\\ {0} & {0} & {2} & {0}\end{array}\right] $$

Find the value(s) of \(h\) for which the vectors are linearly dependent. Justify each answer. \(\left[\begin{array}{r}{1} \\ {5} \\\ {-3}\end{array}\right],\left[\begin{array}{r}{-2} \\ {-9} \\\ {6}\end{array}\right],\left[\begin{array}{r}{3} \\ {h} \\\ {-9}\end{array}\right]\)

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution.

Suppose \(A\) is the \(3 \times 3\) zero matrix (with all zero entries). Describe the solution set of the equation \(A \mathbf{x}=\mathbf{0} .\)

Construct a \(3 \times 3\) nonzero matrix \(A\) such that the vector \(\left[\begin{array}{l}{1} \\ {1} \\ {1}\end{array}\right]\) is a solution of \(A \mathbf{x}=\mathbf{0}\) .