# 9/4/2008

## Number of solutions of a linear system

A system of equations is said to be consistent if there is at least one solution; it is inconsistent if there are no solutions.

A linear system is inconsistent if and only if the reduced row-echelon form of its augmented matrix contains the row [ 0 0 ... 0 | 1], representing the equation 0 = 1.

If a linear system is consistent, then it has either

• infinitely many solutions (if there is at least one free variable), or
• exactly one solution (if all the variables are leading).

## Rank of a matrix

The rank of a matrix A is the number of leading 1s in rref(A).

## Systems with fewer equations than variables

A linear system with fewer equations than unknowns has either no solutions or infinitely many solutions.

## Systems of n equations in n variables

A linear system of n equations in n variables has a unique solution if and only if the rank of its coefficient matrix A is n.

## Matrix Algebra

• Sums of matrices
• Scalar multiples of matrices
• The identity matrix

## The product Ax

• In terms of the columns of A
• In terms of the rows of A
• Algebraic rules

## In class exercises

• Section 1.3 - 6, 9, 11, 19, 21, 29
• Chapter 1 Exercises - 3, 11, 15, 23, 35