MATH1131 Mathematics 1A Algebra

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1 MATH1131 Mthemtics 1A Alger UNSW Sydney Semester 1, 017 Mike Mssierer Mike is pronounced like Mich Plese emil me if you hve ny questions or comments Office hours TBA (week ), or emil me to ook time Office: Red Centre 4105 MATH1131 consists of Lectures: lger, clculus Tutorils: lger, clculus / clssroom, online Clss tests: lger (week 6, 1), clculus (week 5, 9) Mple: online lessons, online quizzes (week 5, 7), l test (week 10) Finl exm (exm period) Get the course notes! (UNSW ookshop, Moodle) 1

2 Moodle Course pck (red the info ook!), Lecturers notes nd slides Announcements, office hours Clssroom tutoril nd homework prolems Links to online tutorils nd ssessments Forum Further support Mthemtics Drop-in Centre (Red Centre 3064) All dministrtive mtters: Student Services Office (Mrkie Lugton) , Red Centre 307 Course overview Chpter 1: Introduction to Vectors (Lectures 1-4) Chpter : Vector Geometry (Lectures 5-8) Chpter 3: Complex Numers (Lectures 9-15) Chpter 4: Liner Equtions nd Mtrices (Lectures 16-0) Chpter 5: Mtrices (Lectures 1-3)

3 Chpter 1: Introduction to Vectors Lecture 1: Vector quntities nd R n Definition Geometriclly, vector is direction nd length (or mgnitude) Algericlly, vector is n ordered set of coordintes These two definitions re equivlent ut offer different viewpoints Consider the vectors nd : 0 the zero vector + ( ) = 3 ( ) 4 = 1 = = ( ) 4 1 ( ) 4 = ( ) 8 + = = ( ) ( ) 4 3 = = ( ) 6 4 ( ) Note tht the position of the vectors is irrelevnt We cn write nd s row vectors or s column vectors In MATH1131, we choose the column vector representtion We cn scle nd y some numer (clled sclr) We cn lso dd nd sutrct nd Geometriclly, vector v is scled y scling its mgnitude v Algericlly, scling, ddition, nd sutrction hppens coordinte-wise 3

4 Definition Loosely, vector spce over R is set of vectors tht cn e dded together nd tht is closed under ddition nd rel sclr multipliction Consider the set V consisting of ll vectors in R nd R 3 ( ) 0 1 This is not vector spce; for instnce, we cnnot dd nd 1 0 { ( ) Consider V = R x } + = R y : x,y > 0 : V This is not vector spce over R under usul + nd : since V is not closed under sclr multipliction The line R, the plne R, nd the spce R 3 re ech vector spces: R R R R 3 the rel line R the rel plne R the rel spce R 3 More generlly, R n is vector spce The complex numers C lso form vector spce (equivlent to R ) 4

5 Definition Formlly, vector spce V over R is set on which ddition + nd sclr multipliction re given so tht, for ll u,v,w V nd λ,µ R, Closure under Addition u+v V Associtive Lw of Addition (u+v)+w = u+(v+w) Commuttive Lw of Addition u+v = v+u Existence of Zero Some element 0 V stisfiesx+0 = x for ll x V Existence of Negtive Some element ( v) V stisfies v+( v) = 0 Closure under Sclr Multipliction λv V Associtive Lw of Sclr Multipliction λ(µv) = (λµ)v Multipliction y identity 1v = v Sclr Distriutive Lw (λ+µ)v = λv+µv Vector Distriutive Lw λ(u+v) = λu+λv Commuttive Lw of Addition Associtive Lw of Addition + + +(+c) + c +c + = + +(+c) = (+)+c 5

6 Simplify 3(+) 3(+) = 3( + ) +( ) (Definition of Sutrction) = (3() + 3) +( ) (Vector Distriutive Lw) = ( (3 )+3 ) +( ) (Associtive Lw of Sclr Multipliction) = ( 6+(+1))+( ) = ( 6+(+1) ) +( ) (Sclr Distriutive Lw) = ( 6+(+) ) +( ) (Multipliction y Identity) = (6+)+ ( +( ) ) (Associtive Lw of Addition) = (6+)+0 (Existence of Negtive) = 6+ (Existence of Zero) In prctice, we just write 3(+) = 6+3 = 6+ Exercise Simplify (4 5) 3( 4) Definition R n is vector spce with entry-wise ddition nd sclr multipliction: x1 + y1 = x 1 +y 1 nd λ x1 = λx 1 x n y n x n +y n λx n Show tht R n stisfies the Commuttive Lw of Addition: u+v = v+u Addition is commuttive in R: u i +v i = v i +u i Thus, u+v = u1 + v1 = u 1 +v 1 = v 1 +u 1 = v+u u n v n u n +v n v n +u n x n 6

7 ( ) 1 Let = nd = ( ) 1 3+ = 3 ( ) 1 4 = 4 ( ) 4 Clculte 3+ nd 4 ( 4 + ( 4 ) ( ) ( ) ( ) ( ) = + = = ) ( ) ( ) ( ) ( ) = = = 8 8 () 9 Exercise 0 Let = nd = 4 Clculte + nd

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