91影视

Row reduce the matrices in Exercise 4 to reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns.

4. \(\left[ {\begin{array}{*{20}{c}}1&3&5&7\\3&5&7&9\\5&7&9&1\end{array}} \right]\)

In Exercise 1-10, assume that\(T\)is a linear transformation. Find the standard matrix of\(T\).

\(T:{\mathbb{R}^3} \to {\mathbb{R}^2}\), rotates points (about the origin) through\( - \frac{\pi }{4}\)radians (clockwise).[Hint:\(T\left( {{e_1}} \right) = \left[ {\frac{1}{{\sqrt 2 }}, - \frac{1}{{\sqrt 2 }}} \right]\)]

In Exercises 1-4, determine if the system has a nontrivial solution. Try to use a few row operations as possible.

4. \(\begin{aligned}{c} - 5{x_1} + 7{x_2} + 9{x_3} = 0\\{x_1} - 2{x_2} + 6{x_3} = 0\end{aligned}\)

Solve each system in Exercises 1鈥�4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure.

4. Find the point of intersection of the lines \({x_1} - 5{x_2} = 1\) and on the line \(3{x_1} - 7{x_2} = 5\).

Compute the products in Exercises 1鈥�4 using (a) the definition, as

in Example 1, and (b) the row鈥搗ector rule for computing \(A{\bf{x}}\). If a product is undefined, explain why.

4. \(\left[ {\begin{array}{*{20}{c}}8&3&{ - 4}\\5&1&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}1\\1\\1\end{array}} \right]\)

The Cambridge Diet supplies .8 g of calcium per day, in addition to the nutrients listed in following table. The amounts of calcium per unit (100 g) supplied by the three ingredients in the Cambridge Diet are as follows: 1.26 g from nonfat milk, .19 g from soy flour, and .8 g from whey. Another ingredient in the diet mixture is isolated soy protein, which provides the following nutrients in each unit: 80 g of protein, 0 g of carbohydrate, 3.4 g of fat, and .18 g of calcium.

1. Set up a matrix equation whose solution determines the amounts of nonfat milk, soy flour, whey, and isolated soy protein necessary to supply the precise amounts of protein, carbohydrate, fat, and calcium in the Cambridge Diet. State what the variables in the equation represent.
2. (M) Solve the equation in (a) and discuss your answer.

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

Consider two vectors ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$ and${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$in R3 that are not parallel.

Which vectors inlocalid="1668167992227" ${\mathbf{鈩�}}^{\mathbf{3}}$are linear combinations of${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}$and${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}$? Describe the set of these vectors geometrically. Include a sketch in your answer.

In Exercises 7-12, use Example 6 to list the eigenvalues of \({\bf{A}}\). In each case, the transformation \({\bf{x}} \mapsto A{\bf{x}}\) is the composition of a rotation and a scaling. Give the angle \(\varphi \) of the rotation, where \( - \pi < \varphi < \pi \), and give the scale factor \(r\).

\(\left( {\begin{aligned}0&{}&{.3}\\{ - .3}&{}&0\end{aligned}} \right)\)

If Ais a $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix with eigenvalues 3 and 4 and if localid="1668109698541" $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$ is a unit eigenvector of A, then the length of vector Alocalid="1668109419151" $\stackrel{\mathbf{鈫�}}{\mathbf{u}}$cannot exceed 4.