Summary
Highlights
The lecture begins by introducing elimination as the primary method for solving systems of linear equations, contrasting it with determinants which will be covered later. The process aims to transform the system matrix into an upper triangular form (U) by systematically eliminating variables. The success or failure of elimination depends on whether a matrix has non-zero pivots.
The lecturer demonstrates the elimination process using a specific 3x3 system of equations (Ax=b). The first pivot is identified as the entry in the top-left corner. Multipliers are determined to create zeros below the pivot in the first column. The process continues to the next pivot, creating an upper triangular matrix (U). The right-hand side (b) is carried along as manipulations are applied to the equations.
The lecture discusses how elimination can fail. A temporary failure occurs if a pivot position is zero, which can often be resolved by swapping rows (row exchange). A complete failure, indicating a non-invertible matrix, happens if a zero appears in a pivot position and there are no non-zero entries below it to swap with.
After converting the original system to an upper triangular form (Ux=c) through elimination, back substitution is used to find the values of the variables. This involves solving the last equation for the last variable, then substituting that value into the second to last equation, and so on, working backward to find all variables.
The lecture transitions to expressing elimination steps using matrix language. Matrix multiplication on the left side of a matrix performs row operations. An elementary matrix (E) is introduced, which represents a single elimination step, such as subtracting a multiple of one row from another. The identity matrix (I) is also discussed as a matrix that performs no operations.
Multiple elimination steps can be combined by multiplying their respective elementary matrices. The associative law of matrix multiplication (E32 * (E21 * A) = (E32 * E21) * A) allows for the grouping of these matrices, meaning a single matrix can be formed to perform all elimination steps at once, transforming A into U.
A permutation matrix (P) is introduced as a type of elementary matrix used to exchange rows. It is formed by swapping rows of the identity matrix. The lecture also clarifies that left multiplication performs row operations, while right multiplication performs column operations.
The concept of inverse matrices is briefly introduced. An inverse matrix (E^-1) is one that undoes the operation of another matrix (E). For an elimination matrix that subtracts a multiple of one row from another, its inverse adds that multiple back. This prepares for future discussions on how to get back from U to A.