t Hei I Y taxi L Y D L A teethed to get the matrix I T A Nettled Use theformula same Theorem 1) Identity transformation:

Size: px

Start display at page:

Download "t Hei I Y taxi L Y D L A teethed to get the matrix I T A Nettled Use theformula same Theorem 1) Identity transformation:"

Audrey Wiggins
5 years ago
Views:

1 ) Identity transformation: Use theformula Theorem to get the matrix I T A Nettled taxi t Hei I Y 2) stretch by factor of.5 horizontally, and.75 vertically: T L A teethed L Y D same

2 3) reflect across the x axis: t K TT o i 4) project onto the y axis: w Tt D A Thistles L it I too

3 5) rotate by an angle 2 (radians) about the origin: A Tk Tied I aaa fo to it okb 6) rotate by an angle about the origin: III I oh I µ lsino.ws gdy.qywgo To cosasino A Neil T I

4 7) horizontal shear with strength.4 : 8) vertical shear with strength -.3:

5 9) mystery linear transformation: ) another mystery linear transformation: A fi B f l s 3 f2, D O

6 Tues Sept.8-.9 linear transformations, and the matrix of a linear transformation. Announcements: We'll discuss today's notes first then go back to Monday examples Warm-up Exercise: So a we use the standard basis vectors I Ez In in IR to understand matrix transformations aka linear transformations express as a linear combination of 5 ej.es in IR Il cs so 35 7 E GET b What's the easiest rector to multiply the matrix I by to yield the 3rd column f Tej in 24

7 Recall, Definition: A function T which has domain equal to n and whose range lies in m is called a linear transformation if it transforms sums to sums, and scalar multiples to scalar multiples. Precisely, T : n m is linear if and only if i T u v = T u T v u, v n ii T c u = c T u c, u n. iii T J 8 whye.g apply Remark: By using the linear transformation properties (i), (ii) multiple times one can verify that linear functions transform linear combinations into linear combinations, with the same weights. For example, H Let's check step by step that if T is linear, then T c u c 2 u 2 c 3 u 3 = c T u c 2 T u 2 c 3 T u 3. why i Tlc ii quiz Tights i T cit TICE ttcgu3 i again c Thi t t ETI This ii lii with c o Thou OTH G TLE Recall as well, In n we write e = :, e 2 = :,... e n = :. We call e, e 2,... e n the standard basis for n because each x x = x 2 : x n n is easily and uniquely expressed as a linear combination of the standard basis vectors, x = x e x 2 e x n e n

8 we did these as warm up exercises Exercise Express as a linear combination of the standard basis vectors. Exercise 2 What vector should we multiply this matrix below by, so that the result is the third column of the matrix? A =

9 Yesterday we showed matrix transformations are linear This is Theorem: Every linear transformation T : n m is actually a matrix transformation, T x = A x, where the j th column of A m n, is T e j, j =, 2.,... n. In other words the matrix of T is A = T e T e... T e 22 h the converse proof: T t Tf x I xie t then x TCEI t xzt E t t intent Tie Ted Tg see.97 Hw A

10 Exercise 3 Illustrate the linear transformation theorem with the projection function T : 3 2, by writing T as a matrix transformation. T x x 2 x 3 = x x 2 formula dotproductury theorem way to 9 8 µ A filet Neil Neil Hoole p S't p A D ft 9 8

11 Exercise 4 Find a matrix formula for the linear transformation T : 2 3 which satisfies T 2 = 2 T =. And sketch the Fmapping diagram. I A filet Neil L o

12 When you learned about functions in a previous course, the following were key ideas: Definitions: The function f : X Y is one to one (sometimes called injective) if each image point of f arises from exactly one input value. In other words, f x = f x 2 x = x 2. The function f : X Y is onto (sometimes called surjective) if for each y Y there is at least one x X so that f x = y. If f : X Y is a function, then the function g : Y X is called an (the) inverse function for f if and only if g f x = x for all x X and f g y = y for all y Y. Theorem A function f : X Y has an inverse function g = f if and only f is one to one and onto. Exercise 5 For a linear function T : n m with matrix A i.e. T x = A x 5a What is the pivot condition on the matrix A that makes the linear function T one to one? 5b What is the pivot condition on the matrix A that makes the function T onto? 5c Is the transformation in Exercise 3 one to one? Is it onto? 5d Is the transformation in Exercise 4 one to one? is it onto?

Math Week 1 notes

Math Week 1 notes Math 2270-004 Week notes We will not necessarily finish the material from a given day's notes on that day. Or on an amazing day we may get farther than I've predicted. We may also add or subtract some