Engineering Analysis

1. Engineering Analysis Chapter 4 Linear Transformations

2. Linear mappings from one vector space to another play an important role in mathematics. This chapter provides an introduction to the theory of such mappings

3. 4.1 Definition and Examples In the study of vector spaces, the most important types of mappings are linear transformations. Definition If L is a linear transformation mapping a vector space V into a vector space W, then it follows from (1) that

4. 4.1 Definition and Examples Conversely, if L satisfies (2) and (3), then Thus, L is a linear transformation if and only if L satisfies (2) and (3).

5. 4.1 Definition and Examples • A linear transformation is a function T that maps a vector space V into another vector space W: V: the domain of T W: the co-domain of T Two axioms of linear transformations

6. Addition in V Addition in W Scalar multiplication in V Scalar multiplication in W 4.1 Definition and Examples • Notes: (1) A linear transformation is said to be operation preserving. (2) A linear transformation from a vector space into itself is called a linear operator.

7. 4.1 Definition and Examples EXAMPLE 1 Let L be the operator defined by L (x) = 3x for each x ∈ R2. Since L (αx) = 3(αx) = α(3x) = αL (x) and L (x + y) = 3(x + y) = 3x + 3y = L (x) + L (y) it follows that L is a linear operator. We can think of L as a stretching by a factor of 3 (see Figure 4.1.1). In general, if α is a positive scalar, the linear operator F(x) = αx can be thought of as a stretching or shrinking by a factor of α.

8. 4.1 Definition and Examples EXAMPLE 2 Consider the mapping L defined by L (x) = x1e1 and it follows that Hence, L is a linear operator. We can think of L as a projection onto the x1-axis (see Figure 4.1.2).

9. 4.1 Definition and Examples EXAMPLE 3 Let L be the operator defined by L (x) = (, − it follows that L is a linear operator. The operator L has the effect of reflecting vectors about the x1-axis (see Figure 4.1.3).

10. 4.1 Definition and Examples EXAMPLE 4 The operator L defined by is linear, since The operator L has the effect of rotating each vector in R2 by 90◦ in the counterclockwise direction (see Figure 4.1.4).

11. 4.1 Definition and Examples EXAMPLE 5 The mapping L: R2 → R1 defined by is a linear transformation, since EXAMPLE 6 Consider the mapping M defined by Since it follows that

12. 4.1 Definition and Examples EXAMPLE 7 The mapping L from R2 to R3 defined by And Note that if we define the matrix A by then for each x ∈ R2.

13. 4.1 Definition and Examples • Ex: Verifying a linear transformation T from R2 into R2 Pf:

14. 4.1 Definition and Examples Therefore, T is a linear transformation.

15. 4.1 Definition and Examples • Ex: Functions that are not linear transformations

16. 4.1 Definition and Examples • Ex: (Linear transformations and bases) Let be a linear transformation such that Find T(2, 3, -2). Sol: (T is a L.T.)

17. (1, 0) (0.5, 0) (0, 1) (0, 0.5) Scale by .5

18. y y x x Scaling by .5

19. φ sin(φ) cos(φ) φ Rotation -sin(φ) cos(φ)

20. y y x x Rotation φ

21. Reflection in y-axis

22. y x Reflection in y-axis y x

23. Reflection in x-axis

24. Reflection in x-axis y y x x

25. Linear Transformations • Scale, Reflection, Rotation, and Shear are all linear transformations • They satisfy: T(au + bv) = aT(u) + bT(v) • u and v are vectors • a and b are scalars • If T is a linear transformation • T((0, 0)) = (0, 0)

26. y y φ x x Rotation about an Arbitrary Point φ This is not a linear transformation. The origin moves.

27. Translation (x, y)(x+a,y+b) y y (a, b) x x This is not a linear transformation. The origin moves.

28. 4.2 Matrix Representations of Linear Transformations In this section, we will see that, for each linear transformation L mapping Rn into Rm, there is an m × n matrix A such that L (x) = Ax If

29. 4.2 Matrix Representations of Linear Transformations We have established that each linear transformation from Rn into Rm can be represented in terms of an m×n matrix. EXAMPLE 1 Define the linear transformation L: R3 → R2 by for each x = (x1, x2, x3in R3. It is easily verified that L is a linear operator. We wish to find a matrix A such that L (x) = Ax for each x ∈ R3. To do this, we must calculate L (e1), L (e2), and L (e3):

30. 4.2 Matrix Representations of Linear Transformations We choose these vectors to be the columns of the matrix

31. 4.2 Matrix Representations of Linear Transformations To check the result, we compute Ax: EXAMPLE 2 Let L be the linear transformation operator R2 that rotates each vector by an angle θ in the counterclockwise direction. We can see from Figure 4.2.1(a) that e1 is mapped into (cos θ, sin θ and the image of e2 is (−sin θ, cos θ . The matrix A representing the transformation will have (cos θ, sin θ as its first column and (−sin θ, cos θas its second column.

32. 4.2 Matrix Representations of Linear Transformations If x is any vector in R2, then, to rotate x counterclockwise by an angle θ, we simply multiply by A [see Figure 4.2.1(b)].

33. y y x x Rotation φ

34. 4.2 Matrix Representations of Linear Transformations EXAMPLE 3 Let L be the linear transformation mapping R3 into R2 defined by L (x) = x1b1 + (x2 + x3)b2 for each x ∈ R3, where Solution The ith column of A is determined by the coordinates of L (ei) with respect to {b1, b2} for i = 1, 2, 3. Thus

35. 4.2 Matrix Representations of Linear Transformations EXAMPLE 4 Let L be a linear transformation mapping R2 into itself defined by Find the matrix A representing L with respect to {b1, b2}. Solution Thus

36. 4.2 Matrix Representations of Linear Transformations EXAMPLE 5 The linear transformation D defined by D(p) = maps P3 into P2. Given the ordered bases [, x, 1] and [x, 1] for P3 and P2, respectively, we wish to determine a matrix representation for D. To do this, we apply D to each of the basis elements of P3.

37. 4.2 Matrix Representations of Linear Transformations Thus,

38. 4.2 Matrix Representations of Linear Transformations Example

39. 4.2 Matrix Representations of Linear Transformations

40. 4.2 Matrix Representations of Linear Transformations

41. 4.2 Matrix Representations of Linear Transformations

42. 4.2 Matrix Representations of Linear Transformations

43. 4.2 Matrix Representations of Linear Transformations

44. 4.2 Matrix Representations of Linear Transformations

45. 4.2 Matrix Representations of Linear Transformations

46. 4.2 Matrix Representations of Linear Transformations

47. 4.2 Matrix Representations of Linear Transformations To find the matrix representation A for a linear transformation L: Rn → Rm with respect to the ordered bases E = {u1, . . . , un} and F = {b1, . . . , bm}, we must represent each vector L (uj) as a linear combination of b1, . . . , bm. The following theorem shows that determining this representation of L (uj) is equivalent to solving the linear system Bx= L (uj). Theorem 4.2.3 Let E = {u1, . . . , un} and F = {b1, . . . , bm} be ordered bases for Rn and Rm, respectively. If L: Rn → Rm is a linear transformation and A is the matrix representing L with respect to E and F, then

48. 4.2 Matrix Representations of Linear Transformations One consequence of this theorem is that we can determine the matrix representation of the transformation by computing the reduced row echelon form of an augmented matrix. The following corollary shows how this is done: Corollary 4.2.4 If A is the matrix representing the linear transformation L : Rn → Rm with respect to the bases

49. 4.2 Matrix Representations of Linear Transformations EXAMPLE 6 Let L: R2 → R3 be the linear transformation defined by Find the matrix representations of L with respect to the ordered bases {u1, u2} and {b1, b2, b3}, where and Solution We must compute L (u1) and L (u2) and then transform the augmented matrix (b1, b2, b3 | L (u1), L (u2)) to reduced row echelon form:

50. 4.2 Matrix Representations of Linear Transformations The matrix representing L with respect to the given ordered bases is