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Chapter 6 Linear Transformations

Chapter 6 Linear Transformations. 6.1 Introduction to Linear Transformations 6.2 Matrices for Linear Transformations 6.3 Similarity. 6.1 Introduction to Linear Transformations. Function T that maps a vector space V into a vector space W :. V : the domain of T W : the codomain of T.

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Chapter 6 Linear Transformations

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  1. Chapter 6Linear Transformations 6.1 Introduction to Linear Transformations 6.2 Matrices for Linear Transformations 6.3 Similarity

  2. 6.1 Introduction to Linear Transformations • Function T that maps a vector space V into a vector space W: V: the domain of T W: the codomain of T

  3. Image of v under T: If v is in V and w is in W such that Then w is called the image of v under T . • the range of T: The set of all images of vectors in V. • the preimage of w: The set of all v in V such that T(v)=w.

  4. Ex 1: (A function from R2 intoR2 ) (a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11) Sol: Thus {(3, 4)} is the preimage of w=(-1, 11).

  5. Linear Transformation (L.T.):

  6. Addition in V Addition in W Scalar multiplication in V Scalar multiplication in W • 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. Ex 2: (Verifying a linear transformation T from R2 into R2) Pf:

  8. Therefore, T is a linear transformation.

  9. Ex 3: (Functions that are not linear transformations)

  10. Notes: Two uses of the term “linear”. (1) is called a linear function because its graph is a line. (2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication.

  11. Zero transformation: • Identity transformation: • Thm 6.1: (Properties of linear transformations)

  12. Ex 4: (Linear transformations and bases) Let be a linear transformation such that Find T(2, 3, -2). Sol: (T is a L.T.)

  13. Ex 5: (A linear transformation defined by a matrix) The function is defined as Sol: (vector addition) (scalar multiplication)

  14. Thm 6.2: (The linear transformation given by a matrix) Let A be an mn matrix. The function T defined by is a linear transformation from Rn into Rm. • Note:

  15. Ex 7: (Rotation in the plane) Show that the L.T. given by the matrix has the property that it rotates every vector in R2 counterclockwise about the origin through the angle . Sol: (polar coordinates) r： the length of v ：the angle from the positive x-axis counterclockwise to the vector v

  16. r：the length of T(v)  +：the angle from the positive x-axis counterclockwise to the vector T(v) Thus, T(v) is the vector that results from rotating the vector v counterclockwise through the angle .

  17. Ex 8: (A projection in R3) The linear transformation is given by is called a projection in R3.

  18. Ex 9: (A linear transformation from Mmn into Mnm ) Show that T is a linear transformation. Sol: Therefore, T is a linear transformation from Mmn into Mnm.

  19. Keywords in Section 6.1: • function: دالة • domain: مجال • codomain:مجال مقابل • image of v under T:صورة تحت التحويل • range of T: T مدى • preimage of w: • linear transformation:تحويل خطي • linear operator:مؤثر خطي • zero transformation:تحويل صفري • identity transformation:تحويل محايد T v

  20. 6.2 Matrices for Linear Transformations • Two representations of the linear transformation T:R3→R3 : • Three reasons for matrix representation of a linear transformation: • It is simpler to write. • It is simpler to read. • It is more easily adapted for computer use.

  21. Thm 6.10: (Standard matrix for a linear transformation)

  22. Pf:

  23. Ex 1: (Finding the standard matrix of a linear transformation) Sol: Vector Notation Matrix Notation

  24. Check: • Note:

  25. Ex 2: (Finding the standard matrix of a linear transformation) Sol: • Notes: (1) The standard matrix for the zero transformation from Rn into Rm is the mn zero matrix. (2) The standard matrix for the zero transformation from Rn into Rn is the nn identity matrix In

  26. Ex 3: (The standard matrix of a composition) Sol:

  27. Inverse linear transformation: • Note: If the transformation T is invertible, then the inverse is unique and denoted by T–1 .

  28. Ex 5: (Finding a matrix relative to nonstandard bases) Sol:

  29. Ex 6: Sol: • Check:

  30. Keywords in Section 6.2: • standard matrix for T:المصفوفة الأساسية للتحويل الخطي • composition of linear transformations:تركيب للتحويلات الخطية • inverse linear transformation:تحويل خطي عكسي • matrix of T relative to the bases B and B‘ : مصفوفة التحويل الخطي T بالنسبة للأساس B و B’ • matrix of T relative to the basis B: مصفوفة التحويل الخطي T بالنسبة للأساس B T

  31. 6.3 Similarity • Ex 4: (Similar matrices)

  32. Ex 5: (A comparison of two matrices for a linear transformation) Sol:

  33. Notes: Computational advantages of diagonal matrices:

  34. Keywords in Section 6.3: • matrix of T relative to B: مصفوفة التحويل الخطي T بالنسبة للأساس B • matrix of T relative to B' : مصفوفة التحويل الخطي T بالنسبة للأساس B’ • transition matrix from B' to B : مصفوفة الانتقال من الأساس B’ الى B • transition matrix from B to B' : مصفوفة الانتقال من الأساس B الى B’ • similar matrix: مصفوفة مماثلة

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