MA 106: Linear Algebra Instructors: Prof. Akhil Ranjan and Prof. S. Krishnan January 4, 2017 1 / 42
Text and references Main Text: E. Kreyszig, Advanced Engineering Mathematics, 8 th ed. (Chapters 6 and 7) Additional references: 1) Notes by Prof. I.K. Rana 2) S. Kumaresan, Linear Algebra- A geometric approach. 3) Wylie and Barrett, Advanced Engineering Mathematics, 6th ed. (Chapter 13) January 4, 2017 2 / 42
More course details Syllabus: see Math Dept webpage. Grading policy: Study hours: 1 Lectures - 21 hours? 2 Tutorials - 7 hours 3 Independent study - 42 hours January 4, 2017 3 / 42
Outline of Week-1 1 Matrices 2 Addition, multiplication, transposition 3 Linear transformations and matrices 4 Linear equations and Gauss’ elimination 5 Row echelon forms and elementary row matrices 6 Reduced REF 7 Gauss-Jordan method for finding inverse January 4, 2017 4 / 42
Matrices Definition 1 A rectangular array of numbers, real or complex, is called a matrix. An array could be of any type of non mathematical objects too. Or more sophisticated mathematical objects like functions instead of numbers. E.g. � sin x � − cos x . cos x sin x Most of the topics today (sec 6.1 and 6.2) will be briefly reviewed and the details will be left for self study January 4, 2017 5 / 42
Basic Notation A = [ a jk ] , 1 ≤ j ≤ m , 1 ≤ k ≤ n denotes an m × n matrix whose entry in j th row and the k th column is the number a jk . (Equivalently, k th entry in the j th row or equivalently j th · · · .) a 11 · · · a 1 k · · · a 1 n . . . . . . . . . a j 1 · · · a jk · · · a jn A = . . . . . . . . . a m 1 · · · a mk · · · a mn January 4, 2017 6 / 42
Examples Some special cases are sometimes named differently Definition 2 An m × 1 matrix is referred to as a column vector while a 1 × n matrix is referred to as a row vector . Examples : 1 is a column. 0 1 − 1 2 [0 1 − 1 3 0] is a row. Two matrices are said to be equal if and only if their corresponding entries are same. January 4, 2017 7 / 42
Transposition Definition 3 A matrix B is a transpose of A if the rows of B are the columns of A and vice versa. Thus, is A = [ a jk ] is an m × n then B is n × m matrix [ b rs ] where b rs = a sr ; 1 ≤ r ≤ n , 1 ≤ s ≤ m . The transpose of A is unique and is denoted by A T . 5 4 � 5 � − 8 1 ⇒ A T = . Example: A = = − 8 0 4 0 0 1 0 Exercise: Show that ( A T ) T = A . January 4, 2017 8 / 42
Symmetry, Addition, Scalar multiplication Definition 4 A matrix A is called symmetric (resp. skew-symmetric) if A = A T (resp. A = − A T ). These are necessarily square matrices i.e. no. of rows=no. of columns. Let A , B be real (or complex) matrices and λ ∈ R (or C ) be a scalar. Definition 5 (Addition) If A = [ a jk ] and B = [ b jk ] have the same order m × n , we define their addition to be A + B = [ c jk ] = [ a jk + b jk ]. Definition 6 (Scalar multiplication) The scalar multiplication of λ with A is defined as λ A = [ λ a jk ]. January 4, 2017 9 / 42
Matrix multiplication If A = [ a jk ] is m × n and B = [ b k ℓ ] is n × p , then the product C := AB is a well defined m × p matrix cooked by the following recipe (called row by column multiplication ): n � C = [ c j ℓ ] where c j ℓ = a jk b k ℓ ; 1 ≤ j ≤ m , 1 ≤ ℓ ≤ p . k =1 b 1 ℓ . . . a j 1 · · · a jk · · · a jn b k ℓ . . . b n ℓ January 4, 2017 10 / 42
Associativity, Distributivity Theorem 7 (Associativity) If A , B , C are real (or complex) matrices such that A is m × n, B is n × p and C is p × q, then the products AB and BC are defined and in turn the products A ( BC ) and ( AB ) C are also defined and the latter two are equal. In other words: A ( BC ) = ( AB ) C Proof: Exercise. Theorem 8 (Distributivity) If B , C are real (or complex) m × n matrices, then A ( B + C ) = AB + AC if A is p × m . ( B + C ) A = BA + CA if A is n × q . Proof: Exercise. January 4, 2017 11 / 42
Transpose of a product Theorem 9 Let A be m × n and B be n × p, then AB and B T A T are well defined and in fact ( AB ) T = B T A T . Proof: Omitted. 4 9 � 3 � 7 , B = . Compute A T , B T , AB , ( AB ) T , Exercise: Let A = 0 2 2 8 1 6 B T A T and A T B T to verify the claim. January 4, 2017 12 / 42
Example: Dot product as a matrix product Definition 10 v 1 w 1 . . . . Let v = and w = be column vectors of the same size n . Their . . v n w n dot product (or inner product or scalar product) is defined as n � v · w = v j w j . j =1 It is interesting to observe that v · w = v T w as a 1 × 1 matrix, which being symmetric also equals w T v = w · v . Question: What about vw T ? Is it defined? Is it also the dot product? [1.0] January 4, 2017 13 / 42
Linear transformations and associated matrices From now on the elements of R n will be written as the column vectors of length n . Definition 11 A map f : R n − → R is called linear if it is of the form f ( x ) = a 1 x 1 + a 2 x 2 · · · + a n x n for suitable constants a 1 , a 2 , ..., a n . Here x 1 , x 2 , ..., x n are the entries of x usually called variables . If we view A = [ a 1 a 2 · · · a n ] as a row vector, then f ( x ) = A x in terms of matrix multiplication. More generally, an m × n matrix A can be viewed as a linear map R n − → R m via x �→ A x . This viewpoint allows us to study matrices geometrically. January 4, 2017 14 / 42
Linear transformations and associated matrices Formal definition of a linear map (also called transformation) Definition 12 A map f : R n − → R m is called linear if it satisfies (i) f ( x + y ) = f ( x ) + f ( y ) and (ii) f ( λ x ) = λ f ( x ) Remark: For an m × n matrix A , f ( x ) = A x is an example of a linear map. These are no other examples of linear maps. A is called the matrix associated to the linear transformation f . � 1 � → R 2 is a Example 1: Show that the range of as a linear map R − − 1 line through 0 . � 1 � t � � � x ( t ) � A ( t ) = [ t ] = = , say. Thus x ( t ) = t , y ( t ) = − t which − 1 − t y ( t ) are parametric equations of the line x + y = 0 through 0 . January 4, 2017 15 / 42
Linear transformations and associated matrices 1 − 1 Example 2:Determine domain and range and also show that − 1 2 0 1 has a plane through 0 as its range. (i) The matrix is 3 × 2, so that’s a linear transformation R 2 − → R 3 . 1 − 1 u − v x ( u , v ) � u � � u � = (ii) B = − 1 2 = − u + 2 v y ( u , v ) , say. v v 0 1 v z ( u , v ) • Thus x = u − v , y = 2 v − u , z = v are the parametric equations of a plane through 0 . • On eliminating u , v the equation of the plane is x + y − z = 0 . January 4, 2017 16 / 42
Linear transformations and associated matrices � 1 1 � Example 3:Consider the matrix: . Determine the images of 0 1 (i) Unit square { 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } , (ii) Unit circle { x 2 + y 2 = 1 } and (iii) Unit disc { x 2 + y 2 ≤ 1 } . (i) Unit square { 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } has vertices �� 0 � � 1 � � 1 � � 0 �� . Therefore the image has vertices , , , 0 0 1 1 �� 0 � � 1 � � 2 � � 1 �� respectively. The full image is a parallelogram , , , 0 0 1 1 with these vertices. � x � � x + y � � u � � x � � u − v � (ii) A = = = ⇒ = and y y v y v x 2 + y 2 = 1 = ⇒ u 2 − 2 uv + 2 v 2 = 1 which is an ellipse. (iii) Elliptic disc u 2 − 2 uv + 2 v 2 ≤ 1 enclosed by the ellipse u 2 − 2 uv + 2 v 2 = 1. January 4, 2017 17 / 42
Linear transformations and associated matrices � 1 � 1 Example 4: Consider the matrix: Determine the images of 1 1 (i) Unit square { 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } , (ii) Unit circle { x 2 + y 2 = 1 } and (iii) Unit disc { x 2 + y 2 ≤ 1 } . (i) Unit square { 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } has vertices �� 0 � � 1 � � 1 � � 0 �� . Therefore the image has ’vertices’ , , , 0 0 1 1 �� 0 � � 1 � � 2 � � 1 �� , , , which are collinear. The full image is a line 0 1 2 1 � 0 � � 2 � segment to . 0 2 � x � � x + y � � u � (ii) A = = and y x + y v √ √ x 2 + y 2 = 1 = ⇒ min ( x + y ) = − 2 , max ( x + y ) = 2 = ⇒ a line � √ � √ � � 2 2 √ √ segment from − to . 2 2 (iii) the line segment above in (ii). [1.5] January 4, 2017 18 / 42
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