In each case, write x as a linear combination of the orthogonal basis of the subspace U. (a)

$$\begin{array} { l } { \mathbf { x } = ( 13 , - 20,15 ) } \\ { U = \operatorname { span } \{ ( 1 , - 2,3 ) , ( - 1,1,1 ) \} } \end{array}$$

, (b)

$$\begin{array} { l } { \mathbf { x } = ( 14,1 , - 8,5 ) } \\ { U = \operatorname { span } \{ ( 2 , - 1,0,3 ) , ( 2,1 , - 2 , - 1 ) \} } \end{array}$$

In each case, show that the set of vectors is orthogonal in

$$\mathbb { R } ^ { 4 }$$

. (a) {(1, -1, 2, 5), (4, 1, 1, -1), (-7, 28, 5, 5)}, (b) {(2, -1, 4, 5), (0, -1, 1, -1), (0, 3, 2, -1)}

In each case find bases for the row and column spaces of A and determine the rank of A. (a)

$$A = \left[ \begin{array} { c c c c } { 2 } & { - 4 } & { 6 } & { 8 } \\ { 2 } & { - 1 } & { 3 } & { 2 } \\ { 4 } & { - 5 } & { 9 } & { 10 } \\ { 0 } & { - 1 } & { 1 } & { 2 } \end{array} \right]$$

, (b)

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

, (c)

$$A = \left[ \begin{array} { r r r r r } { 1 } & { - 1 } & { 5 } & { - 2 } & { 2 } \\ { 2 } & { - 2 } & { - 2 } & { 5 } & { 1 } \\ { 0 } & { 0 } & { - 12 } & { 9 } & { - 3 } \\ { - 1 } & { 1 } & { 7 } & { - 7 } & { 1 } \end{array} \right]$$

, (d)

$$A = \left[ \begin{array} { r r r r } { 1 } & { 2 } & { - 1 } & { 3 } \\ { - 3 } & { - 6 } & { 3 } & { - 2 } \end{array} \right]$$

Miley and Damian determined formulas they could use to compute the first $n-$terms of the series Show that both representations for $S_{n}$ are equivalent.

Show that T is a matrix transformation. (b) T(x1, x2, x3) = (x1, x3, x1 + x2)

We often write vectors in

$$\mathbb { R } ^ { n }$$

as row n-tuples. 1. Obtain orthonormal bases of

$$\mathbb { R } ^ { 3 }$$

by normalizing the following. (a) {(1, -1, 2), (0, 2, 1), (5, 1, -2)}, (b) {(1, 1, 1), (4, 1, -5), (2, -3, 1)}

Find the exact value of each quadrantal angle. If any are not defined, write undefined. $\sin \left( - 1530 ^ { \circ } \right)$

Find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y= cos x. y = cos x/5

In each case find a basis of the subspace U. (a)

$$U = \operatorname { span } \{ ( 1 , - 1,0,3 ) , ( 2,1,5,1 ) , ( 4 , - 2,5,7 ) \}$$

, (b)

$$U = \operatorname { span } \{ ( 1 , - 1,2,5,1 ) , ( 3,1,4,2,7 ) ( 1,1,0,0,0 ) , ( 5,1,6,7,8 ) \}$$

, (c) U=

$$\operatorname { span } \left\{ \left[ \begin{array} { l } { 1 } \\ { 1 } \\ { 0 } \\ { 0 } \end{array} \right] , \left[ \begin{array} { l } { 0 } \\ { 0 } \\ { 1 } \\ { 1 } \end{array} \right] , \left[ \begin{array} { l } { 1 } \\ { 0 } \\ { 1 } \\ { 0 } \end{array} \right] , \left[ \begin{array} { l } { 0 } \\ { 1 } \\ { 0 } \\ { 1 } \end{array} \right] \right\}$$

, (d)

$$U = \operatorname { span } \left\{ \left[ \begin{array} { r } { 1 } \\ { 5 } \\ { - 6 } \end{array} \right] , \left[ \begin{array} { r } { 2 } \\ { 6 } \\ { - 8 } \end{array} \right] , \left[ \begin{array} { r } { 3 } \\ { 7 } \\ { - 10 } \end{array} \right] , \left[ \begin{array} { r } { 4 } \\ { 8 } \\ { 12 } \end{array} \right] \right\}$$

Let {x, y, z, w} be an independent set in

$$\mathbb { R } ^ { n }$$

Which of the following sets is independent? Support your answer. (a) {x-y, y-z, z-x}, (b) {x+y, y+z, z+x}, (c) {x-y, y-z, z-w, w-x}, (d) {x+y, y+z, z+w, w+x}

Besides anxiety, how might you realize that you are suffering from a somatoform or dissociative disorder?

If ||x||=3, ||y||=1, and

$$\mathbf { x } \cdot \mathbf { y } = - 2$$

, compute: (a) ||3x-5y||, (b) ||2x+7y||, (c)

$$( 3 \mathbf { x } - \mathbf { y } ) \cdot ( 2 \mathbf { y } - \mathbf { x } )$$

, (d)

$$( x - 2 y ) \cdot ( 3 x + 5 y )$$

If A is a square matrix, show that det A=0 if and only if some row is a linear combination of the others.

Solve each equation and check your solution. $\frac{6}{5} x=-1$