Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone hentzel/class.307.ICN

Text: Linear Algebra With Applications, Second Edition Otto Bretscher Monday, Jan 27 Chapter 2.1 Page 48 Problems 32,34,42 Main Idea: Matrices can stretch things out and twist them around. Key Words: Linear Transformation, Domain, Codomain, Rotation. Goal: Set up the matrix which does a Linear Transformation.

New Material: | a | | a+b+c | f | b | = | 2a + 3c| | c | | 3b - c | Express f as a matrix. Simply put the necessary coefficients into the matrix A so that you get the correct output. | || a | | a+b+c | | || b | | 2a + 3c | | || c | | 3b - c |

One has 3 piles of sand. The numbered arrows tell what fraction of the pile is given to the other piles. Write out the transition matrix. 0.4 > A B \ < 0.3 / \ /\ /\ / 0.2 \ / 0.5 \/ \ / \/ \ / C

Anew = 0.3 B C A Bnew = 0.4 A C B Cnew = 0.5 B A C | Anew | | | | A | | Bnew | = | | | B | | Cnew | | | | C | Notice that the column sums all equal 1. This means that no material is lost or gained, the matrix just redistributes the material among the three boxes.

Write a matrix that flips elements on the xy plane about the x-axis. | (x,y) | /|\ | | |----- | | | \|/ | (x,-y) | x | ----> | x | | y | | -y | A = | 1 0| | 0 -1|

Write a matrix that flips elements on the xy plane about the y-axis. | (-x,y) (x,y) | | | x | -----> | -x | | y | | y | A = | -1 0 | | 0 1 |

Write a matrix that rotates elements on the xy plane 90 degrees counter clockwise. |. | | 1 | ----> | 0 | | 0 | | 1 | | 0 | ----> | -1 | | 1 | | 0 | | x | ----> | -y | | y | | x | A = | 0 -1 |. | 1 0 |

Write a matrix that rotates an element through an angle theta in the counter clockwise direction. | | /. |/ t

| 1 | ----> | Cos[t] | | 0 | | Sin[t] | | 0 | ----> |-Sin[t] | | 1 | | Cos[t] | | x | ----> | x Cos[t] -y Sin[t] | | y | | x Sin[t] y Cos[t] | A = | Cos[t] - Sin[t] | | Sin[t] Cos[t] |

Application: Rotate the square {0,0},{0,2},{2,2},{2,0} Through an angle of 30 degrees. A = | Sqrt[3]/2 -1/2 | | 1/2 Sqrt[3]/2 | The new corners are {0,0}, {-1,Sqrt[3]}, {Sqrt[3]-1,Sqrt[3]+1}, {Sqrt[3],1}

We can do the same thing in three dimensions. We can project three dimensional points onto the xy plane with a matrix. A = | -1/2 1 0 | | -1/2 0 1 | A is a 2x3 matrix. A point is a 3-tuple which is a 3x1 matrix. Applying A to the point gives a 2x1 matrix which is a point in the xy plane.

What is the image of the unit cube {0,0,0} {0,1,0} {1,1,0} {1,0,0} {0,0,1} {0,1,1} {1,1,1} {1,0,1}

An easy way to set up a matrix is to find out where you want these vectors to go and use that as the matrix. | 1 | | 0 | | 0 | | 0 | | 0 | | 1 | | 0 | | 0 | | 0 | | 0 | | 1 | | 0 | | 0 | | 0 | | 0 | | 1 |

Notice that | 0 | | | | | | | |. | | | | | C1 C2 C3. Cn | | 0 | = | Ci | | | | | | | |. | | | | | 1 |. | 0 |

Explain what this linear transformation does. | | | | | | | 1 | | 1 | It sen ds | 0 | to | 4 |. | 0 | | 7 | | 0 | | 2 | It sends | 1 | to | 5 |. | 0 | | 8 | | 0 | | 3 | It sends | 0 | to | 6 |. | 1 | | 9 |