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Elementary Linear Algebra

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ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 10th February 2010 Comments to the author at [email protected] Contents 1 LINEAR EQUATIONS 1 1.1 Introduction to linear equations . . . . . . . . . . . . . . . . . 1 1.2 Solving linear equations . . . . . . . . . . . . . . . . . . . . . 5 1.3 The Gauss?Jordan algorithm . . . . . . . . . . . . . . . . . . 8 1.4 Systematic solution of linear systems. . . . . . . . . . . . . . 9 1.5 Homogeneous systems . . . . . . . . . . . . . . . . . . . . . . 16 1.6 PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 MATRICES 23 2.1 Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 23

How Haters Help

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R i d e n o u r - B l a d e How Haters Help The Power to Live Your Dreams Thank you for purchasing my book! After you read it , be sure to send your review to lee.r [email protected] for a chance to have it appear on our s ite. I have authored a handful of other works, including: Ways & Means of Time Management Developing Insane Reflexes Decaf: Breaking Your Caffeine Addiction Starting Your Day the Right Way Positively Passionate People Eight Ways to Instantly Calm Yourself Six Ways to Increase Your Energy All Day My works up unti l this point have been primari ly focused on the science of improving your dai ly Li fe; within i t , I have covered a variety of topics ranging from the body to the mind. However, my works have al l shared one problem:

Cross Product

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Calculus
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Cross Product

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Before the geometry, Determinant ? Determinant of a 2? 2 matrix, ? ? ? ? a b c d ? ? ? ? = ad ? bc ? Determinant of a 3? 3 matrix, ? ? ? ? ? ? a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ? ? ? ? ? ? = a 1 ? ? ? ? b 2 b 3 c 2 c 3 ? ? ? ? ? a 2 ? ? ? ? b 1 b 3 c 1 c 3 ? ? ? ? + a 3 ? ? ? ? b 1 b 2 c 1 c 2 ? ? ? ? =a 1 (b 2 c 3 ? b 3 c 2 )? a 2 (b 1 c 3 ? b 3 c 1 ) + a 3 (b 1 c 2 ? b 2 c 1 ) Example 1 1. ? ? ? ? ?1 2 3 5 ? ? ? ? = ?5? 6 = ?11 2. ? ? ? ? ? ? 2 4 6 ?1 3 5 7 2 6 ? ? ? ? ? ? = 2 ? ? ? ? 3 5 2 6 ? ? ? ? ? 4 ? ?

Cross Product

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Cross Product

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Before the geometry, Determinant ? Determinant of a 2? 2 matrix, ? ? ? ? a b c d ? ? ? ? = ad ? bc ? Determinant of a 3? 3 matrix, ? ? ? ? ? ? a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ? ? ? ? ? ? = a 1 ? ? ? ? b 2 b 3 c 2 c 3 ? ? ? ? ? a 2 ? ? ? ? b 1 b 3 c 1 c 3 ? ? ? ? + a 3 ? ? ? ? b 1 b 2 c 1 c 2 ? ? ? ? =a 1 (b 2 c 3 ? b 3 c 2 )? a 2 (b 1 c 3 ? b 3 c 1 ) + a 3 (b 1 c 2 ? b 2 c 1 ) Example 1 1. ? ? ? ? ?1 2 3 5 ? ? ? ? = ?5? 6 = ?11 2. ? ? ? ? ? ? 2 4 6 ?1 3 5 7 2 6 ? ? ? ? ? ? = 2 ? ? ? ? 3 5 2 6 ? ? ? ? ? 4 ? ?

Dot Product (and Vector Projection)

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Dot Product
Vector Projection

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Dot Product Take 3-D for example, De?nition If two vectors ?a =< a 1 , a 2 , a 3 > and ?b =< b 1 , b 2 , b 3 >, then the dot product (inner product, scalar product) of ?a and ?b is de?ned as ? a ? ? b = a 1 b 1 + a 2 b 2 + a 3 b 3 Example < 2,?4 > ? < 3, 5 > = ?14 (?i + 2?j ? 4?k) ? (?2?i +?j + 3?k) = ?12 Remark: Dot product gives a scalar. Properties: 1. ?a ??a = a2 1 + a2 2 + a2 3 = |?a|2 2. ?a ? ?b = a 1 b 1 + a 2 b 2 + a 3 b 3 = ?b ??a 3. ?a ? (?b + ?c) = ?a ? ?b +?a ? ?c 4. (c?a) ? ?b = c(a 1 b 1 + a 2 b 2 + a 3 b 3 ) = c(?a ? ?b) = ?a ? (c?b) 5. ?0 ??a = 0

Organic Chem Review

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Molecular Geometry: VSEPR model stand for valence-shell electron-pair repulsion and predicts the 3D shape of molecules that are formed in bonding. Sigma and Pi Bonds: All single bonds are sigma(?), that occur in the overlap of hybridized orbitals. Pi (?) bonds occur when parallel, unhybridized p orbitals overlap. Double bonds contain one sigma and one pi bond; triple bonds contain one sigma and two pi bonds. ? bonds are weaker than the ? bonds, but because ? bonds are found with ? bonds they are stronger than a single ? bond. Pi bonds also prevent rotation about the bond. Hybridization: Blending of outer bonding orbitals Intermolecular Forces: London Dispersion - Weak intermolecular force, temporary attractive force that results when

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