Linear Algebra Linear Transformations

2 Real Valued Functions Formula Example Description Function from R to R Function from to R Function from to R Function from to R

3 Linear Transformations To illustarte one important way in which transformations can arise, suppose that are real-valued functions of n real variables, say These m equations assign a unique point in to each point in.

4 Introduction Another way to view : Matrix A is an object acting on x by multiplication to produce a new vector Ax or b. Example Axb=

5 Introduction Solving Ax = b amounts to finding all vectors x in R n that are transformed into the vector b in R m under the action of multiplication by A m×n. The correspondence from x to b is a function from one set of vectors to another. This concept is generalization of f:R -> R. x b Multiplication by A =>

6 Introduction A transformation (or function or mapping) T from R n to R m is a rule that assigns to each vector x in R n, a unique vector T(x) in R m. The vector T(x) is called the image of x under T. The set R n is called the Domain of T, and the set of all images T(x) in R m is called Range. We write T:R n -> R m to indicate that the domain of T is R n and the range of T is contained in R m.

7 Introduction

8 Matrix Transformation For each x in R n, Tx is computed as Ax, where A is an m×n matrix. We sometimes denote this Matrix Transformation by x -> Ax. Domain of T lies in R n when A has n columns and range of T lies in R m when each column of A has m entries.

9 Matrix Transformation – Example 1 Example Let and define a transformation T:R 2 ->R 3 by T(x)=Ax, so that

10 a.Find T(u), the image of u under the transformation T b.Find an x in R 2 whose image under T is b. c.Is there more than one x whose image under T is b. d.Determine if c is in the range of the transformation T. Matrix Transformation – Example 1

11 Solution a. b. Solve Tx=b for x. That is, solve Ax=b Matrix Transformation – Example 1

12 Row reduce the augmented matrix Hence The image of this x under T is the given vector b. Matrix Transformation – Example 1

13 c. Any x whose image under T is b must satisfy We have seen in (b) that there is exactly one x whose image is b. Matrix Transformation – Example 1

14 d. The vector c is in the range of T if c is the image of some x in R 2, that is, if c=T(x) for some x. This is just another way of asking if the system Ax=c is consistent. To find the answer, row reduce the augmented matrix The third equation 0=-35 shows that the system is inconsistent. So c is not in the range of T. Matrix Transformation – Example 1

15 Matrix Transformation – Example 2 If Then x -> Ax projects points in R 3 onto R 2. x3x3

16 Matrix Transformation Let Then the transformation T:R 2 -> R 2 defined by Ax is called the Shear Transformation. For example the image of

17 Matrix Transformation Shear Transformation

18 Linear Transformations Theorem If A is an m×n matrix, u and v are vectors in R n and c is a scalar, then a)A(u+v) = Au + Av b)A(cu) = c(Au) Proof For simplicity take n=3, so A=[a 1, a 2, a 3 ]

19 Linear Transformations Similarly (b) can also be proved

20 Linear Transformations A transformation T is linear, if 1. T(u+v) = T(u) + T(v) (for all u,v in the domain of T) 2. T(cu) = cT(u) (for all u and all scalars c) Property (1) says that the result T(u+v) i.e. first adding u+v and then applying T is same as applying T first to u and v and then adding T(u) and T(v).

21 Linear Transformations – Useful Facts If T is a linear transformation 1. T(0) = 0 2. T(cu+dv) = cT(u) + dT(v) For all vectors u, v in the domain of T and all scalars c, d. The general case of (2) is T(c 1 v 1 +…+c p v p ) = c 1 T(v 1 )+…+c p T(v p ) This fact is referred to as Superposition Principle.

22 Linear Transformations – Example 1 Given a scalar r, define T:R 2 -> R 2 by T(x)= rx. T is called a Contraction when and a Dilation when r > 1. Let r = 3, and show that T is a Linear Transformation. Solution Let u, v be in R 2 and let c, d be scalars. Then Obviously T is a Linear Transformation

23 Linear Transformations – Example 1 A dilation transformation

24 Linear Transformations – Example 2 Define a linear transformation T:R 2 -> R 2 by Find the images under T of

25 Linear Transformations – Example 2 Solution Note that T(u+v) is obviously equal to T(u)+T(v). Also, it appears that T rotates u, v and u+v counterclockwise about the origin through 90 degrees.

26 Linear Transformations – Example 2 A rotation transformation

27

Linear transformation

29 Linear Transformation

30 Linear transformation

31 Composition of Transformation

32 Composition of Transformation

33 Composition of Transformation Example:

34 Composition of Transformation

35 Composition of Transformation

36 Linear transformation (Reflections)

37 Linear transformation (Reflections)

38 Orthogonal Projections

39 Orthogonal Projections

40 Orthogonal Projections

41 Contractions & Dilations

42 Rotations

43 Rotations

44 Rotations

45 Example

46 Example (Contd.)

47 Example (Contd.)

48 Example

49 Example (Contd.)

50 Example (Contd.)

51 Example

52 Example

53 Example (Contd.)

54 Example (Contd.)

55 The Matrix of a Linear Transformation Every Linear Transformation T:R n -> R m is a matrix transformation x -> Ax. The key to finding A is to observe that T is completely determined by what it does to the columns of the n×n identity matrix I n.

56 The Matrix of a Linear Transformation – Example 1 The columns of are. Suppose T is a linear transformation from R 2 into R 3 such that With no additional information, find a formula for the image of an arbitrary x in R 2.

57 Solution Write Since T is a linear transformation The Matrix of a Linear Transformation – Example 1

58 The Matrix of a Linear Transformation – Example 1 Thus

59 The Matrix of a Linear Transformation Theorem Let T:R n -> R m be a linear transformation. Then there exist a unique matrix A such that In fact, A is the m×n matrix whose j th column is the vector T(e j ), where e j is the j th column of the identity matrix in R n.

60 Proof Write x=I n x=[e 1 … e n ]x = x 1 e 1 +…+x n e n, and using the Linearity of T to compute The matrix A is called the Standard Matrix for the Linear Transformation T. The Matrix of a Linear Transformation

61 Find the standard matrix A for the dilation transformation T(x) = 3x, for x in R 2. Solution Write The Matrix of a Linear Transformation – Example 1

62 Let T:R 2 -> R 2 be the transformation that rotates each point in R 2 about the origin through an angel, with counterclockwise rotation for a positive angle. Find the standard matrix A of this transformation. Solution rotates to and rotates into The Matrix of a Linear Transformation – Example 2 =>

63 The Matrix of a Linear Transformation – Example 2 A rotation transformation

64 Onto mapping A mapping T:R n -> R m is said to be onto R m if each b in R m is the image of at least one x in R n.

65 Onto Mapping Equivalently, T is onto When the range of T is all the codomain, there exists at least one solution of T(x)=b. The mapping T is not onto when there is some b in for which T(x)=b has no solution.

66 One-to-one mapping A mapping T:R n -> R m is said to be one-to-one if each b in R m is the image of at most one x in R n.

67 One-to-one mapping - Example Let T be the linear transformation whose standard matrix is Solution A happens to be in echelon form, we can see at once that A has a pivot position in each row. For each, the equation Ax=b is consistent. In other words, the linear transformation T maps ( its domain) onto However, since the equation Ax=b has free variables, each b is the image of more than one x, i.e., T is not one-to-one.

68 One-to-one mapping Theorem Let T:R n -> R m be a linear transformation. Then T is one-to-one if and only if the equation T(x)=0 has only the trivial solution.

69 Onto and one-to-one mapping Theorem Let T:R n -> R m be a linear transformation and let A be the standard matrix for T, then: a) T maps R n onto R m if and only if the columns of A span R m b) T is one-to-one if and only if the columns of A are linearly independent Proof a. The columns of A span R m if and only if for each b the equation Ax=b is consistent.

70 Onto and one-to-one mapping In other words, if and only if for every b, the equation T(x)=b has at least one solution. This is true only if and only if T maps R n onto R m. b. The equations T(x)=0 and Ax=0 are similar except notation. T is one-to-one if only if Ax=0 has only the trivial solution. This happens only if the columns of A are linearly independent.

71 Onto and one-to-one mapping - Example Let. Show that T is one-to-one linear transformation. Does T maps R 2 onto R 3. Solution The columns of A are linearly independent. So T is one-to-one.

72 Onto and one-to-one mapping - Example To decide if T is onto R 3, examine the span of the columns of A. Since A is 3×2, the columns of A span R 3 if and only if A has 3 pivot positions. This is impossible, since A has two columns. So the columns of A do not span R 3 and the associated transformation is not onto R 3.

73 Onto and one-to-one mapping - Example Alternatively Let be any typical vector from then must be consistent if T has to be onto.

74 Onto and one-to-one mapping - Example The augmented matrix of the above system is In general is nonzero. Hence the system is inconsistent. Thus T is not onto.

75 Onto and One-to-one Mapping Example The Identity Transformation Let defined by (a) T maps onto since the columns of A span T is one-to-one since the columns of A are linearly independent.