MATH1231_Algebra.pdf - Vector Spaces Linear Transformations Eigenvalues and Eigenvectors Probability and Statistics Supplementary UNSW Mathematics

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Vector Spaces Linear Transformations Eigenvalues and Eigenvectors Probability and Statistics Supplementary UNSW Mathematics Society presents MATH1231/1241 Revision Seminar (Higher) Mathematics 1B Algebra Presented by Joanna Lin and Karen Zhang T2, 2020
Vector Spaces Linear Transformations Eigenvalues and Eigenvectors Probability and Statistics Supplementary Table of Contents I 1 Vector Spaces Vector Spaces Subspaces Linear Combinations and Spans Linear Dependence 2 Linear Transformations Definition of Linear Transformation Linear Maps and Matrices Kernel of a map Image and Rank 3 Eigenvalues and Eigenvectors Definition Computation of eigenvalues and eigenvectors Diagonalisation Applications 2 / 143 MATH1231/1241 Revision Seminar Presented by: Joanna Lin and Karen Zhang
Vector Spaces Linear Transformations Eigenvalues and Eigenvectors Probability and Statistics Supplementary Table of Contents II 4 Probability and Statistics Probability Conditional Probability Random Variables Cumulative Distribution Function Probability Distribution Expected Value and Variance Binomial Distribution Probability Density Function Normal Distribution [X] Exponential Distribution 5 Supplementary More Theory Geometric Representations of Linear Maps Questions 3 / 143 MATH1231/1241 Revision Seminar Presented by: Joanna Lin and Karen Zhang
Vector Spaces Linear Transformations Eigenvalues and Eigenvectors Probability and Statistics Supplementary Vector Spaces 4 / 143 MATH1231/1241 Revision Seminar Presented by: Joanna Lin and Karen Zhang
Vector Spaces Linear Transformations Eigenvalues and Eigenvectors Probability and Statistics Supplementary Vector Spaces Vector Space Definition In the following definition, we assume that u , v and w are elements of a particular vector space. λ and μ are scalars in the field over which the vector space is defined. A vector space V over a field F is a non-empty set in which addition of vectors and scalar multiplication are defined in such a way that the following axioms are satisfied: 1 Closure under Addition . 2 Associative Law of Addition : ( u + v ) + w = u + ( v + w ). 3 Commutative Law of Addition : u + v = v + u . 4 Existence of a Zero : There exists a 0 V such that u + 0 = u . 5 Existence of a Negative : For each u V , there exists a v V where u + v = 0 . 5 / 143 MATH1231/1241 Revision Seminar Presented by: Joanna Lin and Karen Zhang
Vector Spaces Linear Transformations Eigenvalues and Eigenvectors Probability and Statistics Supplementary Vector Space Definition (continued) 6 Closure under Scalar Multiplication . 7 Associative Law of Scalar Multiplication : ( λμ ) u = λ ( μ u ). 8 Multiplication by One : For the scalar 1 F , 1 u = u . 9 Scalar Distributive Law : ( λ + μ ) u = λ u + μ v . 10 Vector Distributive Law : λ ( u + v ) = λ u + λ v . Examples and non-examples of vector spaces Examples of vector spaces include: R n , C n , set of all polynomials and set M n , m of real matrices.