Lesson 1: Systems and Augmented Matrices
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Identify a linear equation and a system of linear equations
- Write a system of linear equations as an augmented matrix
- Perform the three elementary row operations
- Understand the connection between row operations and equation operations
What Is a Linear Equation?
A linear equation in variables x_1, x_2, ..., x_n is an equation that can be written in the form:
where a_1, a_2, ..., a_n and b are real-number constants. The key feature is that each variable appears only to the first power and is not multiplied by another variable.
Linear Equation: An equation where every term is either a constant or a constant times a single variable raised to the first power. No products of variables, no powers, no roots of variables.
Example: Linear vs. Nonlinear
Linear:
- 3x + 2y = 7
- x_1 - 4x_2 + x_3 = 0
- 5x = 10 (one variable is fine)
Not linear:
- x^2 + y = 3 (x is squared)
- xy = 6 (product of two variables)
- sin(x) + y = 1 (nonlinear function of x)
Systems of Linear Equations
A system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
Solution of a System: A list of values (s_1, s_2, ..., s_n) that makes every equation in the system true simultaneously. The solution set is the collection of all solutions.
Example: A 2x2 System
Consider the system:
2x - y = 1
Solution: We need values of x and y that satisfy both equations at the same time.
Try x = 2, y = 3:
- Equation 1: 2 + 3 = 5 ✓
- Equation 2: 2(2) - 3 = 4 - 3 = 1 ✓
Both equations are satisfied, so (2, 3) is the solution.
Example: A 3x3 System
Systems can have any number of equations and variables:
3x_1 - x_2 + 2x_3 = 5
2x_1 + x_2 + x_3 = 7
We need a triple (x_1, x_2, x_3) satisfying all three equations. Solving larger systems by trial and error is impractical, which motivates the matrix methods we will learn.
The Augmented Matrix
Writing out all the variables and plus signs over and over is tedious. We can capture all the essential information (the coefficients and constants) in a compact rectangular array called a matrix.
Augmented Matrix: Given a system of linear equations, the augmented matrix is the matrix formed by writing the coefficients of the variables and the constants in a grid, with a vertical line separating coefficients from constants.
Example: Writing the Augmented Matrix
System:
3x - y + 2z = 5
2x + y + z = 7
Step 1: Line up the coefficients. The coefficient of x in equation 1 is 1, of y is 2, of z is -1. The constant is 4.
Step 2: Write the augmented matrix:
[ 3 -1 2 | 5 ]
[ 2 1 1 | 7 ]
Each row corresponds to one equation. Each column (left of the bar) corresponds to one variable. The column to the right of the bar holds the constants.
Important: Missing Variables
If a variable does not appear in an equation, its coefficient is 0. For example, the system:
2y - z = 1
has augmented matrix:
[ 0 2 -1 | 1 ]
Elementary Row Operations
To solve a system, we manipulate the augmented matrix using operations that change the matrix but do not change the solution set. There are exactly three such operations.
Elementary Row Operations:
- Row Swap (R_i ↔ R_j): Interchange two rows.
- Row Scaling (c R_i → R_i): Multiply every entry in a row by a nonzero constant c.
- Row Replacement (R_i + c R_j → R_i): Add a multiple of one row to another row.
These correspond to operations you already know from algebra:
- Row Swap = reordering equations (doesn't change solutions)
- Row Scaling = multiplying both sides of an equation by a nonzero constant
- Row Replacement = adding a multiple of one equation to another (the elimination step)
Example: Performing Row Operations
Start with:
[ 3 1 | 7 ]
Operation: R_2 - 3R_1 → R_2 (subtract 3 times row 1 from row 2)
New row 2: [3-3(1), 1-3(2) | 7-3(5)] = [0, -5 | -8]
[ 0 -5 | -8 ]
We have created a zero below the first pivot. This is the essence of Gaussian elimination, which we will study fully in Lesson 2.
Example: Row Scaling
Continuing from above, multiply row 2 by -1/5:
(-1/5)R_2 → R_2: [0(-1/5), -5(-1/5) | -8(-1/5)] = [0, 1 | 8/5]
[ 0 1 | 8/5 ]
Now we can read from row 2 that y = 8/5, and from row 1: x + 2(8/5) = 5, so x = 5 - 16/5 = 9/5.
Solution: (x, y) = (9/5, 8/5).
Row Equivalence
Row Equivalent: Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Row-equivalent augmented matrices represent systems with the same solution set.
This is the central idea: we transform a complicated augmented matrix into a simpler one that is easy to solve, knowing the solutions have not changed.
Check Your Understanding
1. Is the equation 3x - 2xy + z = 5 a linear equation? Why or why not?
2. Write the augmented matrix for the system: 2x - 3y = 7 and x + 4y = -1.
3. Starting with the matrix [ 2 4 | 6 ] / [ 1 3 | 5 ], perform R_1 ↔ R_2 (swap the rows). What do you get?
4. Starting with [ 1 3 | 5 ] / [ 2 4 | 6 ], perform R_2 - 2R_1 → R_2. What do you get?
5. In the result [ 1 3 | 5 ] / [ 0 -2 | -4 ], what is the solution to the system?
Key Takeaways
- A linear equation has variables only to the first power with no variable products
- A system of linear equations requires finding values satisfying all equations simultaneously
- The augmented matrix compactly stores all the coefficients and constants
- The three elementary row operations (swap, scale, replacement) transform a matrix without changing its solution set
- Two matrices related by row operations are row equivalent and share the same solutions
Next Steps
Next Lesson
In Lesson 2, you will learn Gaussian Elimination -- the systematic algorithm for using row operations to solve any system of linear equations.
Start Lesson 2