Math 307Spring, 2003Hentzel

Time: 1:10-2:00 MWFRoom: 1324 Howe HallOffice 432 CarverPhone 515-294-8141E-mail: hentzel@iastate.edu

http://www.math.iastate.edu/hentzel/class.307.ICN

Text: Linear Algebra With Applications, Second Edition Otto Bretscher

Wednesday, Jan 29 , Chapter 2.2

Page 61 Problems 2,18,44

Main Idea: Matrices can stretch things out and twist them around.

Key Words: Rotation, Dilation, Linear Transformation

Goal: Look at a matrix and visualize what it does.

Previous Assignment:Page 48 Problems 32,34,42

Page 48 Problem 32:

Find an nxn matrix A such that AX = 3 X

for all X in Rn.

ans: A = 3 I.

Page 48 Problem 34:Consider the transformation T from R2 to R2 that rotates anyvector X through a given angle tin the counterclockwise direction. Find the matrix of T in terms of t.

ans: A = | Cos[t] -Sin[t] | | Sin[t] Cos[t] |.

Page 48 Problem 42:

When you represent a three-dimensional objectgraphically in the plane (on paper, the blackboard, or a computer screen), you have to transform spatial coordinates

| x1 || x2 | into plane coordinates | y1 |. | x3 | | y2 |

The simplest choice is a linear transformation,for example, the one given by the

matrix | -1/2 1 0 | | -1/2 0 1 |.

(a) Use this transformation torepresent the unit cube with cornerpoints | 0 1 0 0 1 0 1 1 | | 0 0 1 0 1 1 0 1 | | 0 0 0 1 0 1 1 1 |

Include the images of the x1, x2, x3axes in your sketch.| -1/2 1 0 | | 0 1 0 0 1 0 1 1 || -1/2 0 1 | | 0 0 1 0 1 1 0 1 | = | 0 0 0 1 0 1 1 1 |

| 0 -1/2 1 0 1/2 1 -1/2 1/2 || 0 -1/2 0 1 -1/2 1 1/2 1/2 |

New Material:

Definition: A linear transformation does just what any reasonable person would expect. These are

(a) T(V+W) = T(V)+T(W) for all vectors V,W.

(b) T(cV) = c T(V) for all numbers c and all vectors V.

If you buy three bags of groceries, the cost of these three bags all together is the same as adding the cost of all three bags separately. Cost[ B1+B2+B3] = Cost[B1]+Cost[B2]+Cost[B3]

The cost of three identical bags of groceries is three times the cost of one of the bags. Cost[3 B] = 3 Cost[ B ].

Now you can argue that if you buy in quantities, then

you can get things cheaper. Fine! Then the cost is not a linear function.

Most processes are linear if the changes are not too large.

For example: If you want to produce 10% more cars,

you need 10% more labor and 10% more material.

In J.I.Case Company in Burlington,they made three models of crawlertractors, the 310, the 750, and the 1000. Their computer had alist of 20,000 parts which were inthe inventory. Whenever an ordercame in for a tractor, the computerwould subtract the parts neededfrom the inventory. As one would suspect, the partsfunction is linear.

If they wanted to make: 3 of the 310, 5 of the 750, 2 of the 1000's, then:Parts(3V310+5V750+2V1000) = 3 Parts(V310) + 5 Parts(V750) + 2 Parts(V1000)

The area of matrices limits itself tothings which behave linearly. Showthat this function is linear. | a | | a+b | T| b | = | b-c |. | c |

First: | 2 | | 5 | What is T | 3 | ? ans |-1 |. | 4 |

| 1 | | 0 | What is T| -1 | ? ans | 0 |. | -1 |

Part (i): We have to check that T(V+W) = T(V) + T(W)

| a | | x | | a+x |T(| b | + | y | ) = T( | b+y | ) | c | | z | | c+z |

= |a+x+b+y| = | a+b | + |x+y| |b+y-c-z| | b-c | |y-z|

| a | | x | = T( | b | ) + T( | y | ). | c | | z |

Part (ii). We have to show that

T(cV) = c T(V).

| x | | cx | T( c | y | ) = T( | cy | ) | z | | cz |

|x|= |cx+cy| = c |x+y|= c T(|y|). |cy–cz| |y-z| |z|

Any linear function can be represented bya matrix. The matrix for T is gotten byevaluating the situation and writing downjust what it has to be. First, Since Tconverts a vector of length three to a vector of length 2, T must be a 2x3 matrix.

| a | | a+b | T| b | = | b - c | | c |

| a | | a + b | T| b | = | b - c | | c |

T = | 1 1 0 | | 0 1 -1 |

Check it with a general vector as follows

| 1 1 0 | | a | | a + b | | 0 1 -1 | | b | = | b - c | | c |

Write the matrix for: | x | | 1x + 2y + z | f| y | = | 2x + 5y - z | | z | | 5x + 4y +24z |Notice that the matrix is just | | 1 | | 0 | | 0 | | | f| 0 | f| 1 | f| 0 | |

| | 0 | | 0 | | 1 | |

| 1 2 1 | = | 2 5 -1 | | 5 4 24 |

(a) Write the matrix for rotation of the x-y plane by 90 degrees.

(b) Write the matrix for rotation of the x-y plane by 45 degrees.

(c) Write the matrix for rotation of the x-y plane by 30 degrees.

What is the inverse for (a).

What is the inverse for (b).

What is the inverse for (c).

Rotation Dilation:

|1 -1| = Sqrt[2]|1/Sqrt[2] -1/Sqrt[2]||1 1| |1/Sqrt[2] 1/Sqrt[2]|

means it rotates through an angle thetawhere Cos[theta] = 1/Sqrt[2] and then stretches it by a factor of Sqrt[2].

Shear: There is a line L such that (1) L is left fixed (2) Things not on L are moved parallel to L.

i.e. T(v) = v for all v on L.

T(v)-v is parallel to L for v not on L.

Show that | 1 1/2 | is a shear. | 0 1 |

| a | ----> | a + 1/2 b | | b | | b |

For | a | to be fixed, b is zero. | b |Therefore the X-axis is fixed.

T| a | - | a | = | 1/2 b | | b | | b | | 0 |Movement is parallel to the X-axis.

Suppose that U and W areperpendicular. That is: U.W = 0.

Show that T[X] = X + (U.X) W is ashear parallel to W.

T[kW] = kW + (U.kW)W = kW since U.W = 0.

T[X]-X = (U.X) W Which is parallel to W.

Projection onto a line in direction U for a Unit vector U.p(X) := (X.U)U. /| X / | / | / | / | / | / | /->------------- <--U.X-->• Line in the direction of U

Reflection in a direction U. X / . / // / ./ / . / //

U/ f(X)

f(X) := 2 (X.U)U - X.f(U) = U If X is perpendicular to U, then Xswitches sign.

The matrix for a rotation is| Cos[t] -Sin[t] || Sin[t] Cos[t] |

The matrix for a reflection is | Cos[t] Sin[t] || Sin[t] -Cos[t] |

The matrix for a dilation is | c 0 | | 0 c |

Express the linear transformation | 6 8 | as a rotation followed | -8 6 | by a dilation. Notice that the matrix is of the form.

10 | 0.6 0.8 | | -0.8 0.6 |Rotation by ArcSin[-0.8] = -53.1301 degreesfollowed by a dilation with factor of 10.

Express the linear transformation| 12 5 || 5 -12 | as a flip followed bya dilation.

13 | 12/13 5/13 | | 5/13 -12/13 |If Cos[theta] = 12/13, this is aflip about the line making theangle 1/2 theta. Then a dilationby the multiple of 13.

How do you tell them apart.

| a -b | | a b |

| b a | | b -a |

rotation reflection

You can view the reflection as

first doing the rotation and then

flipping the new y-axis.