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# LINEAR ALGEBRA MATH 332

Colorado School of Mines

GPA 3.9

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This 13 page Class Notes was uploaded by Diana Prosacco on Monday October 5, 2015. The Class Notes belongs to MATH 332 at Colorado School of Mines taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/219617/math-332-colorado-school-of-mines in Mathematics (M) at Colorado School of Mines.

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Date Created: 10/05/15

MATH 332 Linear Algebra ll MATH332 Linear Algebra Chapter 3 Determinants Section 33 Cramer7s Rule Volume Linear Transformations pgsi 201 209 July 5 2009 Lecture Cramer7s Rule Volume Linear Transformations Cramer s Rule An Inverse Formula Topics Determinants as Volumes Linear Transformations in R2 R3 Prac 1 Problems Prob 5713171925 Section Goals 0 Understand the proof of Cramer s Rule and how this theorem can be used to construct an element level description of inverse of a nonsingular matrix 0 lnterpret the determinant geometrically as the volume of the parallelepiped induced by the vectors Which make up the columns of a matrix Section Objectives 0 Prove Cramer s Rule theorem 37 page 201 through the use of properties of determinants thus giving an element level description of the solution to a square linear system With nonsingular coef cient data 0 Using Cramer s rule and the definition of matrix products to prove theorem 38 Which gives an element level description of an inverse matrix 0 State the implications of theorem 39 Which says that the determinant is related to a unsigned spanned area use this and theorem 310 on page 207 to discuss how the determinant can be used to calculate the way linear transformations change areasi MATH 332 Linear Algebra ll MATH332 Linear Algebra Chapter 05 Linear Equations in Linear Algebra Section 15 Solution Sets of Linear Systems pgsi 5057 June 22 2009 Lecture Solution Sets of Linear Systems Homogeneous systems Topics Nonhomogeneous systems General solution in parametric vector form Prac l 2 Problems Prob 5 15 23 29 31 33 Section Goals 0 Understand how solutions to a nonhomogeneous linear system of equations can be described in terms of V d l t39 to their cor A quot systems 0 A 39 quot and culllctllball character e solutions to linear systems by using a parameter vector form for its description Section Objectives 0 Solve homogeneous and nonhomogeneous linear systems using the row reduction algorithm and report this solution in a parametric vector form that highlights its explicit dependence on the systems freevariables 0 Present and prove theorem 6 on page 53 Which provides a general description of solutions to nonhomogeneous linear systems MATH 332 Linear Algebra MATH332 Linear Algebra Chapter 07 Linear Equations in Linear Algebra Section 17 Linear Independence June 22 2009 pgsi 6572 Lecture Linear Independence Linear Combinations Topics Linear Independence Characterizations of Linearly Dependent Sets P l 4 Problems MC Prob 91517192127 This is one of the most important sections in the text Many other linear algebra textbooks leave this material for much later when it hinders more than it helps The author can do this by initially highlighting the fourfold description of a linear systemi 1 Particularly the vector description found in section 13 pushes the concept of a linear combination to the forefront and this concept can now be used to bread the notion of linear independence The idea of linear independence of vectors is one of the most fundamental concepts in mathe matics A great example is found in differential equations Given dY E AY I has a solution that can be described in terms of the linear combination Yt klvle kQVQeM 2 so long as the eigenvectors V1 V2 are linearly independent 2 This theme is seen in more general settings where the prescription is this 1 Determine what vectorspace7 you are working with3 2 Determine a linearly independent set of vectors which spans the vector space 4 3 Write down arbitrary vectors from this space as linear combinations of the linearly independent vectorsi 1Recall that this is 1 The linear system itself 2 Ax b 3 Alb 4 2 1miai b 2In a more geometric setting we can say that the differential equation de nes a twoedimensional solution space this was what we called phase space and that needs two linearly independent vectors solutions to span it Arbitrary solutions can then be constructed using linear combinations of these basis7 vectors 3A vectorespace is exactly what the name implies a space full of vectors The term space implies that we have a collection of vectors together with some sort of algebra while the term vector can mean quite a lot of things See chapter 4 if you just can7t wait 4Linearly independent vectors are nice but orthogonal vectors are even better One can quickly see that the standard basis vectors of R2 are linearly independent and thus any vector from the plane can take the form x cli 023 but since ij 0 things we will nd are even better MATH 332 Linear Algebra 12 4 Find the weightscoefficientsco ordinates of the linear combination through some sort of al gorithmi 5 Now you have a general technique that can be used to write down7 solutions to problems Well7 at least up to nding some coefficients7 but often much can be said without finding these coefficients 5 Section Goals 0 Understand the connection between linear independence of vectors and trivial solutions to homogeneous equations 0 Characterize linear dependence in terms of linear combinations of vectors Section Objectives 0 De ne linear combination and linear independence of vectors 0 Present and prove theorems 789 from pages 68697 which character e linearly Fl p Fl ht 39 setsi 5We will typically use the rowereduction algorithm but there are cases when this is not a useful tool 6Consider the differential equation y y 0 a quick check of the solution yt k1 sint k2 cost shows the oscillations and frequency of oscillations Finding k1k2 only tells you the amplitudes of oscillation which may not be as important as its other features XL39 2x O 5ch L11 qxlquot sz f GX3O MP BXH clth leago Lmkh d 3 Gun 204 ltl lxiiu cu 1 Tc PLv c D I so AHquot mXOIIMuF H JPN4 Ukuno 57 mx w PKuhO W13 674414 uxuno w Er 43 Q95 9X94 H mungjwrn p mgr 4 wur Abra M WX 4 OTJH gonna LerLlqu Zorn 523 MD o OV PM w Mro ivrv 590993 f IA 93 w V4906 EXMEKJ L X LX353 le 535 9x3 1 Ex Ow oxfu 1 3939 0Q M o gwx 15 Own K1242 Tkli 3 cod 703 m W 4 Arak oh ka ms 3amp3 29651 915 H7 Make sea w 3 a M omos wwulxg avg3amp6 I 95wcv A1 P h 39 A sh Wm M rfMsisi39vd EXt negat 1 Kl BXLS X X73 5 3l XL39gtltBO Eltmgt 1 m3gtltn 5 qu QLt 5X3 1 xtir xzfo EXonvaJ 3 X lxlx XSI39i 39XZ X31 x 3gtlt1 O MATH 332 Linear Algebra MATH332 Linear Algebra Chapter 2 Matrix Algebra Section 21 Matrix Operations pgsi 107118 June 30 2009 Lecture Matrix Operations Sums7 scalar products Matrix Product 1 linear combi of columns 2 row column Toplcs Properties Transpose Prac l7 2 Problems Prob 57 9117 217 23 Section Goals 0 Understand the algebra of matrices and how matrix multiplication relates to composition mapping 0 Understand the operation of matrix transposition and how its properties can be proven using rowcolumn notation at the element leveli Section Objectives 0 De ne matrix multiplication and properties of the noncommutative algebra it forms 0 De ne matrix transposition and prove its properties outlined in theorem 3 MATH 332 Linear Algebra ll MATH332 Linear Algebra Chapter 2 Matrix Algebra Section 22 The Inverse of a Matrix pgsi 118126 June 30 2009 Lecture The Inverse of a Matrix Theorem 4 57 6 Topics Elementary Matrices Finding A 1 Problems Pram 17 2 Prob 81113212335 Section Goals 0 Understand the de nition and properties of a matrix inverse for square data and how this can be used to characterize solutions to Ax b o Devise a method for nding a matrix inverse using elementary rowoperations Section Objectives 0 De ne the inverse matrix for square data and its associated special case for A E R o Prove theorem 57 Which states that for invertible A E Rn there exists a unique solution to Ax b o Prove some of the properties of inverse matrices found in theorem 6 highlighting the change from element level proofs to algebraic proofs on the matrices themselves 0 De ne elementary matrices in connection to rowoperations applied to identity matrices and prove how these matrices can be used to de ne A717 theorem 7 thus giving an algorithm for nding an inverse matrix assuming one exists

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