# MATRICES AND VECTORS SPACE

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1 MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN -SPACE AND -SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR TRANSFORMATIONS DIFFERENTIAL EQUATIONS SYSTEM

2 WEEK Mtrix nd Mtrix s Opertions - Elementry Row Opertions - System of liner equtions - Guss-Jordn Elimintion - Homogenous system - Inverse mtrix - Determinnt - Cofctor expnsion, Row Reduction Methods - Crmmer Method - Lest squre Method - Dot Product, orthogonl projection - Cross product - Spce nd subspce CONTENTS (MID EXAM) August 7, 8 Mtrices nd Mtrix Opertions

3 WEEK Liner independence - Liner Combintion - Bsis nd Dimension - Bsis of Subspce - Bsis of Column spce, Row spce - Inner Product Spce : Norm, ngle nd distnce - Orthogonl nd orthonorml set, projection - Grmm-Schimdt method - Liner Trnsformtion - Trnsformtion mtrix - Kernel nd Rnge of T - Eigen Vlue, Eigen Vector - Digonliztion CONTENTS (FINAL EXAM) - System of Differentil equtions August 7, 8 Mtrices nd Mtrix Opertions

4 MATRICES Mtrix Nottion Definition A mtrix is rectngulr rry of numbers. The numbers in the rry re clled entries in the mtrix. The entry in row i nd column j is denoted by the symbol ij Size of mtrix is described s in terms of the number of rows nd columns it contins K n A Generl m x n mtrix is written s K n M M M M m m K mn August 7, 8 Mtrices nd Mtrix Opertions

5 MATRICES Mtrix Nottion m m n n mn Row Entry row m nd column j Column August 7, 8 Mtrices nd Mtrix Opertions 5

6 August 7, 8 Mtrices nd Mtrix Opertions MATRICES A Squre Mtrix of order n mtrix with n rows nd n columns min digonl :,,, nn 5 B Order Order Min digonl

7 August 7, 8 Mtrices nd Mtrix Opertions 7 MATRICES [ ] I x Identity Mtrices A Squre mtrix with s on the min digonl nd s off the min digonl. Identity mtrix is denoted by I Zero Mtrices A mtrix ll of whose entries re zero I x

8 MATRICES Tringulr Mtrices A Squre mtrix in which ll the entries bove the min digonl re zero (lower tringulr) A Squre mtrix in which ll the entries below the min digonl re zero (upper tringulr) lower tringulr x upper tringulr x August 7, 8 Mtrices nd Mtrix Opertions 8

9 MATRICES Reduced row-echelon form Mtrices Properties of Reduced row-echelon form. If row does not consist entirely of zeros, then the first non-zero number in the row is. We cll this (number ) leding. If there re ny row tht consist entirely of zeros (not-null row), then they grouped together t the bottom of the mtrix. In ny two successive not-null row, the leding in the lower row occurs frther to the right thn the leding in the higher row. Ech column tht contin leding hs zeros everywhere else We will solve system of liner equtions (next chpter) esier when the ugmented mtrices in reduced row echelon form. Mtrix hs only properties, nd is clled hs row-echelon form August 7, 8 Mtrices nd Mtrix Opertions 9

10 August 7, 8 Mtrices nd Mtrix Opertions MATRICES Reduced row-echelon form Mtrices Exmple of reduced row-echelon mtrices Exmple of mtrices not in reduced row-echelon form Properties Properties Properties

11 Opertions on Mtrices Addition nd Subtrction mtrix Sclr Multiples Multiplying Mtrices Trnspose of Mtrix Trce of Mtrix Elementry Row Opertions August 7, 8 Mtrices nd Mtrix Opertions

12 August 7, 8 Mtrices nd Mtrix Opertions Opertions on Mtrices Addition nd Subtrction Definition If A nd B re mtrices of the sme size, then the sum A+B is the mtrix obtined by dding entries of B corresponding entries of A mtrix nd the difference AB is the mtrix obtined by subtrcting entries of B corresponding entries of A mtrix. Exmple : ddition Exmple : subtrction

13 Opertions on Mtrices Sclr Multiples Definition If A is ny mtrix nd k is ny sclr, then the product ka is the mtrix obtined by multiplying ech entry of the mtrix A by k Exmple : sclr multiples August 7, 8 Mtrices nd Mtrix Opertions

14 Opertions on Mtrices Multiplying Mtrices Definition If A is m x r mtrix nd B is r x n mtrix, then the product A x B is the m x n whose entries re determined s follows. The entry in row i nd column j of AB given by formul (AB) ij i b j + i b j + + ir b rj Exmple : multiplying mtrices August 7, 8 Mtrices nd Mtrix Opertions

15 August 7, 8 Mtrices nd Mtrix Opertions 5 Opertions on Mtrices Definition If A is ny m x n mtrix, then the trnspose of A, denoted by A T, is defined to be the n x m mtrix tht the results from interchnging the rows nd columns of A Exmple : trnspose of mtrix 5 A Trnspose of Mtrix 5 T A

16 Opertions on Mtrices Trce of Mtrix Definition If A is n x n squre mtrix, then the trce of A, denoted by tr(a), is defined to be the sum of the entries on the min digonl of A, or given by formul tr(a) nn Exmple : trce of mtrix A tr(a) August 7, 8 Mtrices nd Mtrix Opertions

17 Opertions on Mtrices Elementry Row Opertions (ERO) Elementry row opertions re opertions to eliminte mtrix to be reduced row-echelon form. When we hve the (ugmented mtrix) reduced row-echelon form, we will get solutions of system of liner equtions esier. We will discuss this more t the next chpter There re three types of opertions. Multiply row through by nonzero constnt. Interchnge two rows. Add multiple of one row to nother row August 7, 8 Mtrices nd Mtrix Opertions 7

18 Opertions on Mtrices Steps in elimintion (Crete reduced row-echelon form) We hve mtrix A mxn Go to first row, Chnge entry to be (choose the simplest opertion) Chnge entries,,.., m to be Go to first next row, Chnge entry to be (we pss this step when nd go to the next entry k ) Chnge entries k, k,.., mk to be Repet step nd until the lst row (we will get reduced rowechelon form) August 7, 8 Mtrices nd Mtrix Opertions 8

19 August 7, 8 Mtrices nd Mtrix Opertions 9 Opertions on Mtrices Exmple elimintion using ERO 7 8 A Reduced row echelon form? 7 8 A ~ r r 7 8 ~ r r + 7 Interchnge row nd row * Add - x row to the second row * Leding not leding not zero * we lso cn choose nother opertion

20 August 7, 8 Mtrices nd Mtrix Opertions Opertions on Mtrices Exmple elimintion using ERO () ~ r r + ~ r r + 7 Add - x row to the third row Add - x row to the second row ~ r r + 7 ~ r r + 7 Leding 7 ~ r Add - x row to the first row Add x row to the third row Multiple row by -

21 August 7, 8 Mtrices nd Mtrix Opertions Opertions on Mtrices Exmple elimintion using ERO () Add x row to the second row Add - x row to the first row ~ r r + 7 Leding 7 ~ r r + Reduced row echelon form Notes. All ERO nottions bove is given to help students in understnding this mteril. We don t hve to write this nottions t the next chpter. We cn lso group some ERO nottions to mke shortly

22 Opertion in Mtrices Properties of Mtrix Opertions. A+BB+A b. A+(B+C)(A+B)+C c. A(BC)(AB)C d. A(B+C)AB+AC e. k(ab)(ka)b ; k : sklr f. (A T ) T A g. (AB) T B T A T August 7, 8 Mtrices nd Mtrix Opertions

23 August 7, 8 Mtrices nd Mtrix Opertions Exercises Consider these mtrices B A C D E. Compute the following. BC b. A BC c. CE d. CB e. D CB f. EE T A. Which mtrices below re in reduced row echelon form? 7. b. c. d.

24 August 7, 8 Mtrices nd Mtrix Opertions Exercises. Reduced mtrices below to reduced row echelon form. b. c. 9 d.

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