Announcements September 19

Size: px

Start display at page:

Download "Announcements September 19"

Dwayne Atkinson
5 years ago
Views:

1 Announcements September 19 Please complete the mid-semester CIOS survey this week The first midterm will take place during recitation a week from Friday, September 3 It covers Chapter 1, sections 1 5 and 7 9 Homeworks 15, 17, 18 are due Friday There are three this week so that there can be two next week, the week of the midterm Quiz on Friday: sections 15 and 17 My office hours are Wednesday, 1 2pm and Thursday, 3:3 4:3pm, in Skiles 221 I ll have extra office hours next week As always, TAs office hours are posted on the website Also there are links to other resources like Math Lab

2 Transformations Review from last time Definition A transformation (or function or map) from R n to R m is a rule T that assigns to each vector x in R n a vector T (x) in R m x T R n domain T T (x) range R m codomain (x, sin x) x You may be used to thinking of a function in terms of its graph But for a function from R m to R n, the graph needs m + n dimensions, so it s hard to visualize and draw We won t be graphing most of the transformations in this class

3 Matrix Transformations Most of the transformations we encounter in this class will come from a matrix Definition Let A be an m n matrix The matrix transformation associated to A is the transformation T : R n R m defined by T (x) = Ax In other words, T takes the vector x in R n to the vector Ax in R m The domain of T is R n, which is the number of columns of A The codomain of T is R m, which is the number of rows of A The range of T is the set of all images of T : x 1 T (x) = Ax = v 1 v 2 v n x 2 = x1v1 + x2v2 + + xnvn x n This is the column span of A

4 Matrix Transformations Example 1 1 Let A = 1 and let T (x) = Ax, so T : R 2 R ( ) 1 1 ( ) 7 3 If u = then T (u) = 1 3 = Let b = 5 Find v in R 2 such that T (v) = b Is there more than one? 7 We want to find v such that Av = b We know how to do that: augmented row v = 5 matrix 1 5 reduce ( ) 2 This gives x = 2 and y = 5, or v = (unique) In other words, ( ) 7 T (v) = 1 2 =

5 Matrix Transformations Example, continued 1 1 Let A = 1 and let T (x) = Ax, so T : R 2 R Is there any c in R 3 such that there is more than one w R 2 with T (w) = c? Translation: is there any c in R 3 such that the solution set for Ax = c has more than one vector w in it? The solution set to Ax = b has only one vector v This is a translate of the solution set to Ax = So is the solution set to Ax = c So no! Find c such that there is no v with T (v) = c Translation: Find c such that Ax = c is inconsistent Translation: Find c not in the column span of A (ie, the range of T ) 1 1 a + b We could draw a picture, or notice: a + b 1 = b So 1 1 a + b anything ( in the column span has the same first and last coordinate So 12 ) c = is not in the column span 3

6 Matrix Transformations Geometric example 1 Let A = 1 and let T (x) = Ax, so T : R 3 R 3 x 1 x x T y = 1 y = y z z Then This is projection onto the xy-axis Picture:

7 Matrix Transformations Geometric example Let A = ( ) 1 and let T (x) = Ax, so T : R 2 R 2 1 T ( ) x = y ( 1 1 This is reflection over the y-axis Picture: ) ( ) x = y ( ) x y Then T

8 Let A = ( ) 1 1 and let T (x) = Ax, so T : R 2 R 2 (T is called a shear) 1 Poll What does T do to this sheep? Hint: first draw a picture what it does to the box around the sheep A B C sheared sheep

9 Linear Transformations Recall: If A is a matrix, u, v are vectors, and c is a scalar, then A(u + v) = Au + Av A(cv) = cav So if T (x) = Ax is a matrix transformation then, T (u + v) = T (u) + T (v) T (cv) = ct (v) This property is so special that it has its own name Definition A transformation T : R n R m is linear if it satisfies the above equations for all vectors u, v in R n and all scalars c In other words, T respects addition and scalar multiplication Check: if T is linear, then T () = T (cu + dv) = ct (u) + dt (v) for all vectors u, v and scalars c, d More generally, T ( c 1v 1 + c 2v c nv n ) = c1t (v 1) + c 2T (v 2) + + c nt (v n) In engineering this is called superposition

10 Linear Transformations Dilation Define T : R 2 R 2 by T (x) = 15x Is T linear? Check: T (u + v) = 15(u + v) = 15u + 15v = T (u) + T (v) T (cv) = 15(cv) = c(15v) = c(tv) So T satisfies the two equations, hence T is linear This is called dilation or scaling (by a factor of 15) Picture: T

11 Linear Transformations Rotation Define T : R 2 R 2 by T ( ) x = y ( ) y x Is T linear? Check: (( ) ( )) ( ) ( ) ( ) ( ) u1 v1 u2 v2 (u2 + v 2) u1 + u 2 T + = + = = T u 2 v 2 u 1 v 1 (u 1 + v 1) v 1 + v 2 ( ( )) ( ) ( ) ( ) ( ) v1 cv1 cv2 v2 v1 T c = T = = c = ct v 2 cv 2 cv 1 So T satisfies the two equations, hence T is linear This is called rotation (by 9 ) Picture: v 1 v 2

12 Section 19 The Matrix of a Linear Transformation

13 Linear Transformations are Matrix Transformations Definition The unit coordinate vectors in R n are e 1 = 1, e 2 = 1,, e n 1 = 1, e n = 1 Recall: A matrix A defines a linear transformation T by T (x) = Ax Theorem Let T : R n R m be a linear transformation Let A = T (e 1) T (e 2) T (e n) This is an m n matrix, and T is the matrix transformation for A: T (x) = Ax In particular, every linear transformation is a matrix transformation The matrix A is called the standard matrix for T

14 Linear Transformations are Matrix Transformations Continued Why? Suppose for simplicity that T : R 3 R 2 x 1 T y = T x + y 1 + z z 1 = T ( ) xe 1 + ye 2 + ze 3 = xt (e 1) + yt (e 2) + zt (e 3) x = T (e 1) T (e 2) T (e 3) y z x = A y z So when we think of a matrix as a function from R n to R m, it s the same as thinking of a linear transformation

15 Linear Transformations are Matrix Transformations Example We defined the dilation transformation T : R 2 R 2 by T (x) = 15x What is its standard matrix? ( ) 15 T (e 1) = 15e 1 = ( ) 15 ( ) = A = 15 T (e 2) = 15e 2 = 15 Check: ( ) ( ) x = y ( ) 15x = 15 15y ( ) x = T y ( ) x y

16 Linear Transformations are Matrix Transformations Example Question What is the matrix for the linear transformation T : R 2 R 2 defined by (Check linearity ) T (x) = x rotated counterclockwise by an angle θ? T (e 2) T (e 1) = T (e 2) = T (e 1 ) θ sin(θ) cos(θ) cos(θ) e 1 sin(θ) ( ) cos(θ) sin(θ) ( ) θ = cos(θ) sin(θ) ( 9 = ) ( ) = A = 1 sin(θ) sin(θ) cos(θ) A = 1 cos(θ) θ e 2

17 Linear Transformations are Matrix Transformations Example Question What is the matrix for the linear transformation T : R 3 R 3 that reflects through the xy-plane and then projects onto the yz-plane? e 1 yz reflect xy yz project yz yz xy xy xy T (e 1) =

18 Linear Transformations are Matrix Transformations Example, continued Question What is the matrix for the linear transformation T : R 3 R 3 that reflects through the xy-plane and then projects onto the yz-plane? yz yz yz reflect xy project yz xy e 2 xy xy T (e 2) = e 2 = 1

19 Linear Transformations are Matrix Transformations Example, continued Question What is the matrix for the linear transformation T : R 3 R 3 that reflects through the xy-plane and then projects onto the yz-plane? e 3 yz reflect xy yz project yz yz xy xy xy T (e 3) = 1

20 Linear Transformations are Matrix Transformations Example, continued Question What is the matrix for the linear transformation T : R 3 R 3 that reflects through the xy-plane and then projects onto the yz-plane? T (e 1) = T (e 2) = 1 T (e 1) = 1 = A = 1 1

21 Other Geometric Transformations There is a long list of geometric transformations of R 2 in 19 of Lay (Reflections over the diagonal, contractions and expansions along different axes, shears, projections, ) Please look them over

Announcements Monday, September 25

Announcements Monday, September 25 The midterm will be returned in recitation on Friday. You can pick it up from me in office hours before then. Keep tabs on your grades on Canvas. WeBWorK 1.7 is due Friday