Linear Equations

Thursday, 20 June 2019

2:40 PM

Machine generated alternative text: MATH2099 Musings Lecture 2 - Linear Equations 2019-06-06 Linear Algebra To solve a system of linear equations Aæ = b, we reduce the augmented matrix A 1b to row- echelon form by Gaussian Elimination • A leading entry is the first non-zero entry in a row • Row-echelon form of a matrix satisfies the conditions o Leading entries appear from left to right when progressing down o All zero-rows are at the bottom • A leading column is a column with leading entries Solutions of a System • 0 solutions - if the row-echelon form has a leading right-hand column • 1 solution - all columns on the left are leading • infinite solutions - if there are solutions, and some columns on the left are non-leading Example Details Find the solutions of the system Row 1 : C — y 3 1 1 -1 Upon reducing, we get o 3 1 3 -2 0 0 8-4 c-2 • When c2 — 4 # 0 (ie when c # 4-2) the system has a unique solution Performing back substitution Row 3. c2 — 4)z = c —2 Row 2: y — cz = (62) -2 2(C+1) So we get the solution • When c solution —2, the last row is (0 4) , which means that the system has no When c = 2, the row-echelon form is 3 0 o Set z 1 o -2 So we reach 2 2 1 1 Matrix Algebra • The sum of two matrices is defined if and only if they are of the same size. • If A is of size m x n and B is of size p x q then the product AB exists if and only if n The resulting matrix will have a size of m x q • Unless A = B, then AB # BA (order is important') • A matrix is said to be symetric if AT = A • A matrix is said to be skew-symmetric if AT = —A • If an inverse of a matrix A exists, A is said to be invertable • Else it is singular or non-invertible Properties of Transpose and Inverse • (AA + gB)T = AAT + gBT for all A, B € M,nn and scalars • If AB exists, then (AB)T = BTAT • If A € Mnn is invertible, then the inverse of A is unique • If both A and B in Mnn are invertible, then AB is invertible and (AB)¯I Example Details ,Orove that if A is invertible, then (A-I)T = To prove that X = (A-I)T is the inverse of AT, we need to show that AT X Now.. ATx = = (VIA)T = IT = 1 Example Details Let A, B, C be invertible matrices of the same size. Suppose • A is symmetric • B is skew-symmetric • C is orthogonal (C¯l = CT) Simplify -BB(C IC)B 1B -BBIB-IIC-IB 1 -BBB-IC-IB-I Powered by Hugo I Theme - Even 0 2019 Andrew Wong @5206677) x AT Next @

 

 

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