# lecture22 - Lecture Lecture 22 Revision Lecture I 18 days...

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ecture 22 Lecture 22 Revision Lecture I

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18 days to final exam! Looking ahead • Week 12: • Lecture 22 & 23 (Revision lectures & Exam info) • Tutorial 10 o lab (self practice) No lab (self practice) • Week 13: • No lecture •LabQu iz • Tutorial 11 (download from workbin) eek 14 (reading week): Week 14 (reading week): • Chat room (Friday evening) • Final exam (Saturday 9am) Lecture 22 Announcement 2
Where can you get help? • VT consultation (till April 23) • Email me (till April 23) • Discussion forum (till April 23) • Math Clinic (by graduate students, till Apr 23) ± Monday, 3-5pm ± Tuesday, 3-5pm ± Friday, 1-3pm enue: S16 #05- 9 ± Venue: S16 #05 99 • Counselling and Psychological Services ± Exam Stress Announcement 3

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Summary Concepts from Chapters 1 - 7
Matrix & Associated Terminologies linear verse determinant linear transformation transformation w inverse quare row echelon form transpose Non row echelon form transpose Square Matrix Square Matrix eigenvalue eigenvector char.eqn row sp column sp ullspace rank nullity row sp column sp nullspace rank nullity 5 nullspace

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column space of AB Œ column space of A Matrix multiplication - alternative form m x n " A : m x n ( ) n a a a A 2 1 = x x 2 1 x x : n x 1 x = ? n x # x = " Ax ? " n n x x x a a a Ax + + + 2 2 1 1 a linear combination of the columns of A n x k AB = ? ( ) k b b b B 2 1 ( ) k Ab Ab Ab AB " 2 1 = olumns of B terms of olumns of B : n x k 6 columns of AB in terms of columns of B each column of AB is a linear combination of columns of A
Many faces of linear system Standard form Augmented matrix form = + + + = + + + n n n n b x a x a x a b x a x a x a " " 2 2 2 12 1 21 1 1 2 12 1 11 n n b a a a b a a a " " 2 2 22 21 1 1 12 11 = + + + m n mn m m b x a x a x a " # 2 2 1 1 m mn m m b a a a " # # # # 2 1 (A | b ) n b x a a a " 1 1 1 12 11 n b a a a 1 1 12 11 Matrix equation form Vector equation form = n n mn m m n b b x x a a a a a a # # " # # # " 2 2 2 1 2 22 21 = + + + n n mn n m m b b x a a x a a x a a # # " # # 2 2 2 2 22 1 1 21 7 Ax = b x 1 a 1 + x 2 a 2 + ··· + x n a n = b

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Solutions of linear system Ax = b (with n variables) no solution unique solution homogeneous () no parameter Ax = 0 infinite solutions g system can have 1 to n-1 parameters eneral solution (example) ⎛− 2 3 1 1 x General solution (example) + = t s x 1 3 2 + + = 0 5 0 1 2 0 0 0 1 4 3 2 t s x x x = = = t x s x 3 2 2 5 8 particular solution of Ax = b general solution of Ax = 0 t x 4
There could be alternative methods to solve these problems.

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## This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture22 - Lecture Lecture 22 Revision Lecture I 18 days...

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