Section 1.1 : Systems of Linear Equations

Topics

• Systems of linear equations
• Matrix notation
• Elementary row operations
• Questions of existence and uniqueness of solutions

What is a linear equation?

Form of a linear equation

• , … , and are the coefficients
• , … , are the variables or unknowns
• is the dimension, or number of variables

Essentially, linear equations involve variables, which can be scaled and summed together

For example

• is a line in two dimensions
• is a plane in three dimensions

Analyzing the above two examples

Two dimensions

• Represents a line where you can traverse in two directions, contained in a 2D plane
• Infinite number of solutions, as you can traverse along the line infinitely in either direction
• You can rewrite the

Three dimensions

• In the 3D plane, you are able to traverse in 3 directions
• With one equation, it limits it to 2 directions of movement (degrees of freedom!)

Systems of linear equations

When we have more than one linear equation, we have a linear system of equations. For example, a linear system with two equations is

A system can have a unique solution, no solution, or an infinite of number of solutions

Two-dimensional case

Consider the following systems. How are they different from each other?

The number of solutions differ in each case

• In the parallel lines example, through linear operations, we see that the equations can have the same coefficients but equal to a different constant. There will be no points that satisfy both.
• In the identical lines example, the second equation can be manipulated using linear operations (in this case scale by -1) to yield the exact same equation as the first
  • The line is not linearly independent
  • Similar to having one single line in a 2D plane, there are an infinite number of solutions.

Three-dimensional case

An equation defines a plane in . The solution to a system of three equations is the set of intersections of the planes.

Note: In the infinite solutions case, although there are three planes, the addition of a third plane did not impose an additional restriction

Row Reduction by Elementary Row Operations

How can we find the solution set to a set of linear equations? We can manipulate equations in a linear system using row operations.

1. (Replacement/Addition) Add a multiple of one row to another.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply a row by a non-zero scalar.

Let’s use these operations to solve a system of equations.

Augmented Matrices

It is redundant to write , , again and again, so we rewrite systems using matrices.

For example,

…can be written as the augmented matrix,

Note: Pretend there’s a vertical line to indicate it’s augmented :)

The vertical line reminds us that the first three columns are the coefficients to our variables , , .

Exercise: Identify the solution to the linear system above.

Consistent Systems and Row Equivalence

Definition (Consistent)

• A linear system is consistent if it has at least one solution.

Definition (Row Equivalence)

• Two matrices are row equivalent if a sequence of linear matrix operations transforms one matrix into the other.

Note: If the augmented matrices of two linear systems are row equivalent, then they have the same solution set.

Fundamental questions

Two questions that we will revisit many times throughout our course.

1. Does a given linear system have a solution? In other words, is it consistent?
2. If it is consistent, is the solution unique?

Preview of next lesson (L1.2)

• Formalizing echelon form and RREF
• Leading entry definition

Next lecture: 05.18 Lecture - MATH1554