Lesson 5 Linear Transformations in Geometry
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1 Lesson 5 Linear Transformations in Geometr Math 21b Februar 14, 2007 Announcements Homework for Februar 16: 2.2: 6,8,16,30,34,47*,50* Problem Session: Frida 1-2pm in SC 103b (?) M office hours: Monda 2-4, Tuesda 3-5 in SC 323 Visit the Section Zero blog: k14308&pageid=icb.page70643
2 Last Time Definition A function T from R m to R n is called a linear transformation if there is an n m matri A such that for all in R m. T ( ) = A.
3 Last Time Definition A function T from R m to R n is called a linear transformation if there is an n m matri A such that for all in R m. T ( ) = A. Theorem Let T : R m R n be a function. Then T is a linear transformation if and onl if (a) T ( v + w) = T ( v) + T ( w) (b) T (k v) = kt ( v) for all vectors v and w in R m and scalars k. Another propert (not independent of the first two) is that T ( 0) = 0.
4 Which of these are linear transformations? (a) T () = ([ ]) (b) T = 2 2 sin t (c) γ(t) = cos t t ([ ]) [ ] + (d) T =
5 Let T : R 2 R 2 be the transformation which rotates the plane b 30, then reflects in the -ais.
6 Let T : R 2 R 2 be the transformation which rotates the plane b 30, then reflects in the -ais. Question Is this is a linear transformation?
7 Is T ( v + w) = T ( v) + T ( w)?
8 Is T ( v + w) = T ( v) + T ( w)? v
9 Is T ( v + w) = T ( v) + T ( w)? v
10 Is T ( v + w) = T ( v) + T ( w)? T ( v) v
11 Is T ( v + w) = T ( v) + T ( w)? T ( v) v w
12 Is T ( v + w) = T ( v) + T ( w)? T ( v) v w
13 Is T ( v + w) = T ( v) + T ( w)? T ( v) v w T ( w)
14 Is T ( v + w) = T ( v) + T ( w)? v + w T ( v) v w T ( w)
15 Is T ( v + w) = T ( v) + T ( w)? v + w T ( v) v w T ( w)
16 Is T ( v + w) = T ( v) + T ( w)? T ( v) v v + w w T ( v + w) T ( w)
17 Is T ( v + w) = T ( v) + T ( w)? T ( v) v v + w w T ( v + w) T ( w)
18 Let T : R 2 R 2 be the transformation which rotates the plane b 30, then reflects in the -ais. Question Is this is a linear transformation? Answer. Yes!
19 Let T : R 2 R 2 be the transformation which rotates the plane b 30, then reflects in the -ais. Question Is this is a linear transformation? Answer. Yes! Question What is the matri A representing T?
20 Getting the matri from its effect 1 2 T ( v) = T. = T ( 1 e e m e m ) m = 1 T ( e 1 ) + 2 T ( e 2 ) + + m T ( e m ) = 1 a a m a m 1 = [ ] 2 a 1 a 2 a m. m = A v
21 Upshot To find the matri of a linear transformation, find the image of each standard basis vector and form the matri whose columns are those images.
22 Looking at our original T, we can find the images of each standard basis vector:
23 Looking at our original T, we can find the images of each standard basis vector: [ ] [ ] [ ] 1 3/2 3/2 e 1 = 0 1/2 1/2
24 Looking at our original T, we can find the images of each standard basis vector: [ ] [ ] [ ] 1 3/2 3/2 e 1 = 0 1/2 1/2 [ ] [ ] [ ] 1 1/2 1/2 e 2 = 0 3/2 3/2
25 Looking at our original T, we can find the images of each standard basis vector: [ ] [ ] [ ] 1 3/2 3/2 e 1 = 0 1/2 1/2 [ ] [ ] [ ] 1 1/2 1/2 e 2 = 0 3/2 3/2 So [ 3/2 1/2 A = 1/2 3/2 ]
26 Tpes of Linear Transformations
27 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T.
28 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution
29 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution e 1
30 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution T ( e 1 ) = e 1 ] [ 1/2 0
31 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution e 2 T ( e 1 ) = e 1 ] [ 1/2 0
32 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution [ ] 0 T ( e 2 ) = 1/2 e 2 T ( e 1 ) = e 1 ] [ 1/2 0
33 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution
34 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution e 2 e 1
35 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution [ ] 0 T ( e 2 ) = 1/2 T ( e 1 ) = [ 1/2 0 ]
36 Scaling Eample Let f : R 2 R 2 be the linear transformation which scales the plane b a factor of 1 2. Find the matri representing T. Solution [ ] 0 T ( e 2 ) = 1/2 T ( e 1 ) = [ 1/2 0 ] [ ] 1/2 0 We have A =. 0 1/2
37 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. [ ] 2 0. Describe the 0 1
38 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1
39 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1 e 1
40 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1 e 1 T ( e 1 ) = [ ] 2 0
41 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1 e 2 e 1 T ( e 1 ) = [ ] 2 0
42 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1 [ ] 0 T ( e 2 ) = e 1 2 e 1 T ( e 1 ) = [ ] 2 0
43 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1
44 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1 e 2 e 1
45 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1 [ ] 0 T ( e 2 ) = 1 T ( e 1 ) = [ ] 2 0
46 Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 2 0. Describe the 0 1 [ ] 0 T ( e 2 ) = 1 T ( e 1 ) = [ ] 2 0 T scales horizontal distances b 2 but leaves vertical distances alone.
47 The Dot Product Remember the dot product from Math 21a: v w = v 1 w 1 + v 2 w w m w m. Then (a) v w = w v (b) v ( u + w) = v u + v w (c) (k v) w = k( v w) = v (k w) (d) v v 0 for all v. v v = 0 v = 0. In fact, v = v v (e) v w = 0 v and w are perpendicular
48 Orthogonal Decomposition Fact Let L be an line and an vector. Then can be decomposed as = + where is parallel to L and is perpendicular to L.
49 Orthogonal Decomposition Fact Let L be an line and an vector. Then can be decomposed as = + where is parallel to L and is perpendicular to L. L
50 Orthogonal Decomposition Fact Let L be an line and an vector. Then can be decomposed as = + where is parallel to L and is perpendicular to L. L
51 Proof. Let v be an vector (so v v = 1) parallel to L. Then = k v for some k, and v = 0. So v = ( + ) v = v + v = k( v v) + 0 so k = v v v, meaning = v v v v, =.
52 Projections and Reflections Given L, there are two linear transformations associated to the orthogonal decomposition = + : The projection onto L: The reflection in L: proj L ( ) = ref L ( ) =
53 Eample Let f : R 2 R 2 be the linear transformation which projects onto the line = 2. Find the matri representing T.
54 Eample Let f : R 2 R 2 be the linear transformation which projects onto the line = 2. Find the matri representing T. Solution A = [ 1/5 2/5 2/5 4/5 ]
55 Eample Let f : R 2 R 2 be the linear transformation which reflects in the line = 2. Find the matri representing T.
56 Eample Let f : R 2 R 2 be the linear transformation which reflects in the line = 2. Find the matri representing T. Solution We have [ ] e 1 = e 1 e 4/5 1 = 2 /5 e 2 = e 2 e 2 = [ 2/5 1/5 ] So [ ] T ( e 1 ) = e 1 3/5 e 1 = [ ] 4/5 3/5 4/5 [ ] = A = T ( e 2 ) = e 4/5 2 4/5 3/5 e 2 = 3/5
57 Rotations Eample Let f : R 2 R 2 be the linear transformation which rotates an angle θ. Find the matri representing T.
58 Rotations Eample Let f : R 2 R 2 be the linear transformation which rotates an angle θ. Find the matri representing T. Solution [ ] cos θ sin θ A = sin θ cos θ
59 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. [ ] 1 1/2. Describe the 0 1
60 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1
61 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1 e 1
62 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1 e 1 [ ] 1 T ( e 1 ) = 0
63 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1 e 2 e 1 [ ] 1 T ( e 1 ) = 0
64 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1 e 2 T ( e 2 ) = e 1 [ ] 1 T ( e 1 ) = 0 [ 1/2 1 ]
65 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1
66 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1 e 2 e 1
67 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1 T ( e 1 ) = T ( e 2 ) = [ ] 1 0 [ 1/2 1 ]
![Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2.](/docs-images/60/44732904/images/68-0.png "Describe the 0 1 T ( e 1 ) = T ( e 2 ) = [ ] 1 0 [ 1/2 1 ] T is called a (horizontal) shear transformation.")
68 Shears Eample Let T : R 2 R 2 be multiplication b A = geometric effect of T on the plane. Solution [ ] 1 1/2. Describe the 0 1 T ( e 1 ) = T ( e 2 ) = [ ] 1 0 [ 1/2 1 ] T is called a (horizontal) shear transformation.
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