MA 106 Linear Algebra

Transcription

1 1/44 MA 106 Linear Algebra Sudhir R. Ghorpade January 5, 2016

2 Generalities about the Course 2/44 INSTRUCTORS: Prof. Santanu Dey (D1 & D3) and Prof. Sudhir R. Ghorpade (D2 & D4)

3 Generalities about the Course 2/44 INSTRUCTORS: Prof. Santanu Dey (D1 & D3) and Prof. Sudhir R. Ghorpade (D2 & D4) LECTURES: D2: Mon 9.30, Tue 10.30, Thu D4: Mon, Thu pm, all in LA 102 TUTS: Wed 3-4 pm in LT 004, 005, 006, 105, 106 (for D2) and LT 304, 305, 306, 205, 206 (for D4)

4 Generalities about the Course 2/44 INSTRUCTORS: Prof. Santanu Dey (D1 & D3) and Prof. Sudhir R. Ghorpade (D2 & D4) LECTURES: D2: Mon 9.30, Tue 10.30, Thu D4: Mon, Thu pm, all in LA 102 TUTS: Wed 3-4 pm in LT 004, 005, 006, 105, 106 (for D2) and LT 304, 305, 306, 205, 206 (for D4) OFFICE: 106 B, First Floor, Maths Dept. OFFICE HOURS: Tue, 12-1 pm or by appointment.

5 Generalities about the Course 2/44 INSTRUCTORS: Prof. Santanu Dey (D1 & D3) and Prof. Sudhir R. Ghorpade (D2 & D4) LECTURES: D2: Mon 9.30, Tue 10.30, Thu D4: Mon, Thu pm, all in LA 102 TUTS: Wed 3-4 pm in LT 004, 005, 006, 105, 106 (for D2) and LT 304, 305, 306, 205, 206 (for D4) OFFICE: 106 B, First Floor, Maths Dept. OFFICE HOURS: Tue, 12-1 pm or by appointment. ATTENDANCE: Compulsory! Random name calling in each class. We may not rely on biometric attendance. EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up) + 1 Common Quiz + Final. Marks: = 50.

6 Generalities about the Course 2/44 INSTRUCTORS: Prof. Santanu Dey (D1 & D3) and Prof. Sudhir R. Ghorpade (D2 & D4) LECTURES: D2: Mon 9.30, Tue 10.30, Thu D4: Mon, Thu pm, all in LA 102 TUTS: Wed 3-4 pm in LT 004, 005, 006, 105, 106 (for D2) and LT 304, 305, 306, 205, 206 (for D4) OFFICE: 106 B, First Floor, Maths Dept. OFFICE HOURS: Tue, 12-1 pm or by appointment. ATTENDANCE: Compulsory! Random name calling in each class. We may not rely on biometric attendance. EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up) + 1 Common Quiz + Final. Marks: = 50. BONUS: 2.5 Extra Marks for 100 % Attendance; However, 1 mark for each absentee.

7 Generalities about the Course INSTRUCTORS: Prof. Santanu Dey (D1 & D3) and Prof. Sudhir R. Ghorpade (D2 & D4) LECTURES: D2: Mon 9.30, Tue 10.30, Thu D4: Mon, Thu pm, all in LA 102 TUTS: Wed 3-4 pm in LT 004, 005, 006, 105, 106 (for D2) and LT 304, 305, 306, 205, 206 (for D4) OFFICE: 106 B, First Floor, Maths Dept. OFFICE HOURS: Tue, 12-1 pm or by appointment. ATTENDANCE: Compulsory! Random name calling in each class. We may not rely on biometric attendance. EXAMS: 6 Quizzes in Tuts (Best 5 taken; no make up) + 1 Common Quiz + Final. Marks: = 50. BONUS: 2.5 Extra Marks for 100 % Attendance; However, 1 mark for each absentee. MORE INFO: See the Moodle page of the course or srg/106 2/44

8 What is Linear Algebra 3/44 WIKIPEDIA DESCRIPTION: Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

9 What is Linear Algebra 3/44 WIKIPEDIA DESCRIPTION: Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. MOREOVER: Linear algebra is classically related to the study of: Systems of linear equations and their solutions Matrices Determinants...

10 What is Linear Algebra 3/44 WIKIPEDIA DESCRIPTION: Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. MOREOVER: Linear algebra is classically related to the study of: Systems of linear equations and their solutions Matrices Determinants... For a more specific decription, at least as far as this course is concerned, see the official syllabus.

11 Why Study Linear Algebra 4/44 SHORT ANSWER: Because it is beautiful!

12 Why Study Linear Algebra 4/44 SHORT ANSWER: Because it is beautiful! BUT ALSO BECAUSE IT IS: One of the most important basic areas in all of Mathematics, having an impact comparable to that of Calculus. Provides a vital arena where the interaction of Mathematics and machine computation is seen. Many of the problems studied in Linear Algebra are amenable to systematic and even algorithmic solutions, and this makes them implementable on computers. Many geometric topics are studied making use of concepts from Linear Algebra. Applications to Physics, Engineering, Probability & Statistics, Economics and Biology.

13 Cartesian coordinate space 5/44 R denotes the set of all real numbers

14 Cartesian coordinate space 5/44 R denotes the set of all real numbers The space R n is the totality of all ordered n-tuples (x 1,..., x n ) where x 1,..., x n vary over R. It is called the n-dimensional Euclidean space or the n-dimensional Cartesian coordinate space

15 Cartesian coordinate space 5/44 R denotes the set of all real numbers The space R n is the totality of all ordered n-tuples (x 1,..., x n ) where x 1,..., x n vary over R. It is called the n-dimensional Euclidean space or the n-dimensional Cartesian coordinate space Elements of R n are referred to as vectors when n > 1. Elements of R may be referred to as scalars.

16 Cartesian coordinate space 5/44 R denotes the set of all real numbers The space R n is the totality of all ordered n-tuples (x 1,..., x n ) where x 1,..., x n vary over R. It is called the n-dimensional Euclidean space or the n-dimensional Cartesian coordinate space Elements of R n are referred to as vectors when n > 1. Elements of R may be referred to as scalars. For i = 1,..., n, the function π i : R n R defined by π i ((x 1,..., x n )) = x i is called the i th coordinate function or the i th coordinate projection.

17 Cartesian coordinate space R denotes the set of all real numbers The space R n is the totality of all ordered n-tuples (x 1,..., x n ) where x 1,..., x n vary over R. It is called the n-dimensional Euclidean space or the n-dimensional Cartesian coordinate space Elements of R n are referred to as vectors when n > 1. Elements of R may be referred to as scalars. For i = 1,..., n, the function π i : R n R defined by π i ((x 1,..., x n )) = x i is called the i th coordinate function or the i th coordinate projection. Given a function f : A R n and 1 i n, the function f i : A R defined by f i := π i f is called the i th component function of f. These f i completely determine f. And we may write f = (f 1,..., f n ). 5/44

18 Algebraic structure of R n 6/44 Addition: For x = (x 1,..., x n ), y = (y 1,..., y n ) define x + y = (x 1 + y 1,..., x n + y n )

19 Algebraic structure of R n 6/44 Addition: For x = (x 1,..., x n ), y = (y 1,..., y n ) define x + y = (x 1 + y 1,..., x n + y n ) Usual laws of addition hold. We set: 0 = (0,..., 0), x = ( x 1,..., x n )

20 Algebraic structure of R n 6/44 Addition: For x = (x 1,..., x n ), y = (y 1,..., y n ) define x + y = (x 1 + y 1,..., x n + y n ) Usual laws of addition hold. We set: 0 = (0,..., 0), x = ( x 1,..., x n ) Scalar multiplication: For α R and x R n, define αx := (αx 1,..., αx n ).

21 Algebraic structure of R n 6/44 Addition: For x = (x 1,..., x n ), y = (y 1,..., y n ) define x + y = (x 1 + y 1,..., x n + y n ) Usual laws of addition hold. We set: 0 = (0,..., 0), x = ( x 1,..., x n ) Scalar multiplication: For α R and x R n, define αx := (αx 1,..., αx n ). The following properties clearly hold:

22 Algebraic structure of R n 6/44 Addition: For x = (x 1,..., x n ), y = (y 1,..., y n ) define x + y = (x 1 + y 1,..., x n + y n ) Usual laws of addition hold. We set: 0 = (0,..., 0), x = ( x 1,..., x n ) Scalar multiplication: For α R and x R n, define αx := (αx 1,..., αx n ). The following properties clearly hold: Associativity: α(βx) = (αβ)x for all α, β R, x R n

23 Algebraic structure of R n 6/44 Addition: For x = (x 1,..., x n ), y = (y 1,..., y n ) define x + y = (x 1 + y 1,..., x n + y n ) Usual laws of addition hold. We set: 0 = (0,..., 0), x = ( x 1,..., x n ) Scalar multiplication: For α R and x R n, define αx := (αx 1,..., αx n ). The following properties clearly hold: Associativity: α(βx) = (αβ)x for all α, β R, x R n Distributivity: α(x + y) = αx + αy α R, x, y R n

24 Algebraic structure of R n Addition: For x = (x 1,..., x n ), y = (y 1,..., y n ) define x + y = (x 1 + y 1,..., x n + y n ) Usual laws of addition hold. We set: 0 = (0,..., 0), x = ( x 1,..., x n ) Scalar multiplication: For α R and x R n, define αx := (αx 1,..., αx n ). The following properties clearly hold: Associativity: α(βx) = (αβ)x for all α, β R, x R n Distributivity: α(x + y) = αx + αy α R, x, y R n 1x = x for all x R n 6/44

25 7/44 2. LINEAR MAPS ON EUCLIDEAN SPACES AND MATRICES Definition A map f : R n R m is said to be a linear if f (αx + βy) = αf (x) + βf (y) α, β R, x, y R n. Examples: Projection map π i : R n R; inclusion map R n R n+t ; multiplication by a fixed scalar

26 7/44 2. LINEAR MAPS ON EUCLIDEAN SPACES AND MATRICES Definition A map f : R n R m is said to be a linear if f (αx + βy) = αf (x) + βf (y) α, β R, x, y R n. Examples: Projection map π i : R n R; inclusion map R n R n+t ; multiplication by a fixed scalar dot product by a fixed vector in R n gives a linear map from R n to R; what about the converse?

27 7/44 2. LINEAR MAPS ON EUCLIDEAN SPACES AND MATRICES Definition A map f : R n R m is said to be a linear if f (αx + βy) = αf (x) + βf (y) α, β R, x, y R n. Examples: Projection map π i : R n R; inclusion map R n R n+t ; multiplication by a fixed scalar dot product by a fixed vector in R n gives a linear map from R n to R; what about the converse? f : R n R m linear f i linear for each i = 1,..., m.

28 7/44 2. LINEAR MAPS ON EUCLIDEAN SPACES AND MATRICES Definition A map f : R n R m is said to be a linear if f (αx + βy) = αf (x) + βf (y) α, β R, x, y R n. Examples: Projection map π i : R n R; inclusion map R n R n+t ; multiplication by a fixed scalar dot product by a fixed vector in R n gives a linear map from R n to R; what about the converse? f : R n R m linear f i linear for each i = 1,..., m. Distance travelled is a linear function of time when velocity is constant. So is the voltage as a function of resistance when the current is constant.

29 2. LINEAR MAPS ON EUCLIDEAN SPACES AND MATRICES Definition A map f : R n R m is said to be a linear if f (αx + βy) = αf (x) + βf (y) α, β R, x, y R n. Examples: Projection map π i : R n R; inclusion map R n R n+t ; multiplication by a fixed scalar dot product by a fixed vector in R n gives a linear map from R n to R; what about the converse? f : R n R m linear f i linear for each i = 1,..., m. Distance travelled is a linear function of time when velocity is constant. So is the voltage as a function of resistance when the current is constant. x, x n (n > 1), sin x, etc. are not linear 7/44

30 8/44 Exercise: (i) Show that if f : R n R m is a linear map then k f ( α i x i ) = i=1 k α i f (x i ) x i R n and α i R. i=1

31 8/44 Exercise: (i) Show that if f : R n R m is a linear map then f ( k α i x i ) = i=1 k α i f (x i ) x i R n and α i R. i=1 (ii) Show that the projection on a line L passing through the origin defines a linear map of R 2 to R 2 and its image is equal to L.

32 8/44 Exercise: (i) Show that if f : R n R m is a linear map then f ( k α i x i ) = i=1 k α i f (x i ) x i R n and α i R. i=1 (ii) Show that the projection on a line L passing through the origin defines a linear map of R 2 to R 2 and its image is equal to L. (iii) Show that rotation through a fixed angle θ is a linear map from R 2 R 2.

33 8/44 Exercise: (i) Show that if f : R n R m is a linear map then f ( k α i x i ) = i=1 k α i f (x i ) x i R n and α i R. i=1 (ii) Show that the projection on a line L passing through the origin defines a linear map of R 2 to R 2 and its image is equal to L. (iii) Show that rotation through a fixed angle θ is a linear map from R 2 R 2. (iv) By a rigid motion of R n we mean a map f : R n R n such that d(f (x), f (y)) = d(x, y) x, y R n.

34 8/44 Exercise: (i) Show that if f : R n R m is a linear map then f ( k α i x i ) = i=1 k α i f (x i ) x i R n and α i R. i=1 (ii) Show that the projection on a line L passing through the origin defines a linear map of R 2 to R 2 and its image is equal to L. (iii) Show that rotation through a fixed angle θ is a linear map from R 2 R 2. (iv) By a rigid motion of R n we mean a map f : R n R n such that d(f (x), f (y)) = d(x, y) x, y R n. Show that a rigid motion of R 3 which fixes the origin is a linear map.

35 Structure of linear maps 9/44 L(n, m) = Set of all linear maps from R n to R m

36 Structure of linear maps 9/44 L(n, m) = Set of all linear maps from R n to R m For f, g L(n, m) define αf and f + g by (αf )(x) = αf (x); (f + g)(x) = f (x) + g(x)

37 Structure of linear maps 9/44 L(n, m) = Set of all linear maps from R n to R m For f, g L(n, m) define αf and f + g by (αf )(x) = αf (x); (f + g)(x) = f (x) + g(x) If f L(n, m) and g L(p, n), then f g L(p, m)

38 Structure of linear maps 9/44 L(n, m) = Set of all linear maps from R n to R m For f, g L(n, m) define αf and f + g by (αf )(x) = αf (x); (f + g)(x) = f (x) + g(x) If f L(n, m) and g L(p, n), then f g L(p, m) If f, g L(n, 1), then define fg : R n R by Does fg L(n, 1)? (fg)(x) = f (x)g(x).

39 Structure of linear maps 9/44 L(n, m) = Set of all linear maps from R n to R m For f, g L(n, m) define αf and f + g by (αf )(x) = αf (x); (f + g)(x) = f (x) + g(x) If f L(n, m) and g L(p, n), then f g L(p, m) If f, g L(n, 1), then define fg : R n R by (fg)(x) = f (x)g(x). Does fg L(n, 1)? Let e i = (0,..., 0, 1, 0,..., 0) (standard basis elements). If x R n, then x = n i=1 x ie i.

40 Structure of linear maps 9/44 L(n, m) = Set of all linear maps from R n to R m For f, g L(n, m) define αf and f + g by (αf )(x) = αf (x); (f + g)(x) = f (x) + g(x) If f L(n, m) and g L(p, n), then f g L(p, m) If f, g L(n, 1), then define fg : R n R by (fg)(x) = f (x)g(x). Does fg L(n, 1)? Let e i = (0,..., 0, 1, 0,..., 0) (standard basis elements). If x R n, then x = n i=1 x ie i. If f L(n, m), then f (x) = i x i f (e i )

41 Conversely, if given v 1,..., v n R m, define a (unique) linear map f by assigning f (e i ) = v i 10/44

42 Conversely, if given v 1,..., v n R m, define a (unique) linear map f by assigning f (e i ) = v i Examples: 10/44

43 Conversely, if given v 1,..., v n R m, define a (unique) linear map f by assigning f (e i ) = v i Examples: 1. Given a f L(n, 1), if we put u = (f (e 1 ),..., f (e n )), then for each x R n we can write f (x) = i x if (e i ) = u.x 10/44

44 10/44 Conversely, if given v 1,..., v n R m, define a (unique) linear map f by assigning f (e i ) = v i Examples: 1. Given a f L(n, 1), if we put u = (f (e 1 ),..., f (e n )), then for each x R n we can write f (x) = i x if (e i ) = u.x 2. Consider a system of linear equations a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b a m1 x 1 + a m2 x a mn x n = b m

45 10/44 Conversely, if given v 1,..., v n R m, define a (unique) linear map f by assigning f (e i ) = v i Examples: 1. Given a f L(n, 1), if we put u = (f (e 1 ),..., f (e n )), then for each x R n we can write f (x) = i x if (e i ) = u.x 2. Consider a system of linear equations a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b a m1 x 1 + a m2 x a mn x n = b m The set of all solutions of the j th equation is a hyperplane P j in R n. Solving the system means finding the intersection P 1... P m of these hyperplanes.

46 On the other hand the LHS of each of these equations can be thought of as a linear map T i : R n R, which associates to (x 1,..., x n ) R n the scalar a i1 x 1 + a i2 x a in x n in R. 11/44

47 11/44 On the other hand the LHS of each of these equations can be thought of as a linear map T i : R n R, which associates to (x 1,..., x n ) R n the scalar a i1 x 1 + a i2 x a in x n in R. Together, they define a linear map T L(R n, R m ) given by T = (T 1,..., T m ).

48 On the other hand the LHS of each of these equations can be thought of as a linear map T i : R n R, which associates to (x 1,..., x n ) R n the scalar a i1 x 1 + a i2 x a in x n in R. Together, they define a linear map T L(R n, R m ) given by T = (T 1,..., T m ). Solving the system of linear equations corresponds to determining x R n such that T (x) = b, where b = (b 1,..., b m ) Henceforth, we will think of R n as the collection of column vectors x 1 x 2. = (x 1, x 2,..., x n ) T x n Here the superscript T stands for transpose. 11/44

49 Matrix representation of a linear map 12/44 As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors.

50 Matrix representation of a linear map 12/44 As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors (of size m) viz., f (e 1 ),..., f (e n ). Place them side by side:

51 Matrix representation of a linear map 12/44 As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors (of size m) viz., f (e 1 ),..., f (e n ). Place them side by side: Thus if f (e j ) = (f 1j, f 2j,..., f mj ) T, then we obtain f 11 f f 1n f 21 f f 2n M f =... f m1 f m2... f mn

52 Matrix representation of a linear map 12/44 As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors (of size m) viz., f (e 1 ),..., f (e n ). Place them side by side: Thus if f (e j ) = (f 1j, f 2j,..., f mj ) T, then we obtain f 11 f f 1n f 21 f f 2n M f =... f m1 f m2... f mn This array is called a matrix with m rows and n columns.

53 Matrix representation of a linear map 12/44 As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors (of size m) viz., f (e 1 ),..., f (e n ). Place them side by side: Thus if f (e j ) = (f 1j, f 2j,..., f mj ) T, then we obtain f 11 f f 1n f 21 f f 2n M f =... f m1 f m2... f mn This array is called a matrix with m rows and n columns. We say matrix M f is of size m n.

54 Notation: M f = ((f ij )) 13/44

55 13/44 Notation: M f = ((f ij )) Matrices are equal if their sizes are the same and the entries are the same.

56 13/44 Notation: M f = ((f ij )) Matrices are equal if their sizes are the same and the entries are the same. If m = 1 we get row matrices; column matrices if n = 1 we get

57 13/44 Notation: M f = ((f ij )) Matrices are equal if their sizes are the same and the entries are the same. If m = 1 we get row matrices; column matrices if n = 1 we get M : L(n, m) M m,n f M f is one-one and called matrix representation of linear maps

58 13/44 Notation: M f = ((f ij )) Matrices are equal if their sizes are the same and the entries are the same. If m = 1 we get row matrices; column matrices if n = 1 we get M : L(n, m) M m,n f M f is one-one and called matrix representation of linear maps M f +g = M f + M g ; M αf = αm f