Math 102 Linear Algebra I Stefan Martynkiw These notes are adapted from lecture notes taught by Dr.Alan Thompson and from “Elementary Linear Algebra: 10th Edition” :Howard Anton. Picture above sourced from (http://i.imgur.com/RgmnA.gif) 1/52 Table of Contents Chapter 3 – Euclidean Vector Spaces.........................................................................................................7 3.1 – Vectors in 2-space, 3-space, and n-space......................................................................................7 Theorem 3.1.1 – Algebraic Vector Operations without components...........................................7 Theorem 3.1.2 .............................................................................................................................7 3.2 – Norm, Dot Product, and Distance................................................................................................7 Definition 1 – Norm of a Vector..................................................................................................7 Definition 2 – Distance in Rn......................................................................................................7 Dot Product.......................................................................................................................................8 Definition 3 – Dot Product...........................................................................................................8 Definition 4 – Dot Product, Component by component..............................................................8 Theorems 3.2.2 and 3.2.3.............................................................................................................8 Geometry in Rn.................................................................................................................................9 Theorem 3.2.5 – Triangle Inequality............................................................................................9 Theorem 3.2.6..............................................................................................................................9 Theorem 3.2.7..............................................................................................................................9 Dot Products as Matrix Multiplication.............................................................................................9 3.3 – Orthogonality.........................................................................................................................10 Point-Normal forms of lines and planes.........................................................................................10 Theorem 3.3.1 – Point-Normal Forms of Lines and Planes......................................................10 Orthogonal Projections...................................................................................................................11 Theorem 3.3.3 – Pythagorean Theorem in Rn...........................................................................11 Magnitudes of projections..........................................................................................................11 Distance Problems..........................................................................................................................12 Theorem 3.3.4............................................................................................................................12 3.4 – Geometry of Linear Systems......................................................................................................13 Vector & Parametric Equations of Lines and Planes .....................................................................13 Theorem 3.4.1............................................................................................................................13 Theorem 3.4.2............................................................................................................................13 Dot Product form of a Linear System.............................................................................................13 3.5 – Cross Product.............................................................................................................................14 Theorem 3.5.1 -- Relationships between Cross Product and Dot Product................................14 Theorem 3.5.2 – Properties of Cross Product............................................................................14 Magnitude of a Cross Product....................................................................................................14 Scalar Triple Product..................................................................................................................14 Geometric Interpretation of Determinants......................................................................................14 Theorem 3.5.4............................................................................................................................14 Chapter 4 – General Vector Spaces..........................................................................................................15 4.1 – Real Vector Spaces.....................................................................................................................15 Vector Space Axioms......................................................................................................................15 Definition 1 – Vector Space Axioms..........................................................................................15 Theorem 4.1.1............................................................................................................................15 4.2 – Subspaces...................................................................................................................................16 Definition 1................................................................................................................................16 Theorem 4.2.1............................................................................................................................16 2/52 Building Subspaces.........................................................................................................................16 Theorem 4.2.2............................................................................................................................16 Definition 2................................................................................................................................16 Theorem 4.2.3............................................................................................................................16 Definition 3................................................................................................................................17 Theorem 4.2.4............................................................................................................................17 Theorem 4.2.5............................................................................................................................17 4.3 – Linear Independence..................................................................................................................17 Definition 1................................................................................................................................17 Theorem 4.3.1............................................................................................................................17 Theorem 4.3.2............................................................................................................................18 Geometric Interpretation of Linear Independence..........................................................................18 Theorem 4.3.3............................................................................................................................18 4.4 – Co-ordinates and Basis...............................................................................................................18 Definition 1................................................................................................................................18 Theorem 4.4.1 – Uniqueness of Basis Representation..............................................................18 Definition 2................................................................................................................................19 4.5 – Dimension..................................................................................................................................19 Number of Vectors in a Basis.........................................................................................................19 Theorem 4.5.1 ...........................................................................................................................19 Theorem 4.5.2............................................................................................................................19 Definition 1................................................................................................................................19 Example 1..................................................................................................................................19 Example 2 – Dimension of a solution space..............................................................................20 Plus / Minus Theorem.....................................................................................................................20