Systems of Linear Equations and Ax=b
Outline: 1) The relationship between systems of linear equations and Linear Combinations of vectors 2) The relationship between systems of equations and 3) Basic rules of Matrix  vector products as linear combinations of vectors 4) : the fundamental issues 5) Solving linear systems: Introduction to Gaussian Elimination 
Systems of linear Equations A simple example : x  y = 1 2x + y = 5 Solution by "elimination" and "backsubstitution": 

The geometry of Linear Systems: the "row"
picture

The geometry of Linear Systems:
the "column" picture Linear systems of equations as Linear Combinations of vectors x  y = 1 

The geometry of Linear Systems: the "column" picture  3x3 problems (3 equations, 3 unknowns ):  
A more compact approach: Introducing the Matrix A x  y = 1 2x + y = 5 2x2 problem: 3x3 problem: 
MatrixVector Products:
the Column Picture: Example: 

MatrixVector Products:
the Row Picture (dot products): Example: Point!: Both Column and Row approaches are identical 
Matrix Component Notation: Matrices in Matlab: 

Big Points: 1) is a linear combination of the columns of A 2) is a vector 3) maps to a new vector The Forward Problem: you know A and , just find . Easy! Just use Matrix Vector multiplication . A solution must always exist. The Inverse Problem: Given A and , find , such that Much Harder! This is the problem of solving linear systems. Fundamental Math questions : Given square A and a vector 1) does a solution exist such that ? 2) is the solution unique ? 3) How do you find ? 
The geometry of


The geometry of
: Interesting complications...

General results (to be shown): for square nxn A: has exactly one of three outcomes 1) There is a unique solution 2) There is no solution 3) There are an infinite number of solutions Question: How can you tell and how do you find the solution? 

Gaussian Elimination: A systematic algorithm to both diagnose and solve linear systems Consider our toy 2x2 problem again:
Suppose you wanted to eliminate x instead of y in
2nd equation? 
Gaussian Elimination: A systematic algorithm to transform a general matrix A to upper Triangular form. Upper Triangular system: Easily solved by Back Substitution : 

Gaussian Elimination: A systematic algorithm to transform a general matrix A to upper Triangular form. Need to transform both A and the RHS b to maintain equality. Useful to Consider the Augmented matrix Idea of Elimination: Use row operations to zero out elements below the pivot using multipliers of the pivot row 
Gaussian Elimination: A systematic algorithm to transform a general matrix A to upper Triangular form. Examples: If it "works", Gaussian Elimination transforms A→U with n distinct pivots. n Pivots implies a unique solution. 

Failure of Gaussian Elimination: Consider: 
Temporary Failure of Gaussian
Elimination: Consider: The Fix: Row exchange (permutation operation 

Gaussian Elimination on a 3x3 system of
equations: Consider: 2x + y + 3z = 3 

Gaussian Elimination on a 3x3
system of equations: Example 2 (row exchange): x + 2y + 4z = 1 2x + 4y + 2z = 2 6x + 10y  z = 8 Matrix Vector form: Augmented Matrix form: 

Gaussian Elimination: The Overall Pattern  Linear combinations in 3 Dimensions


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