Matrix Form of Linear Systems

Hello and welcome to your first lesson in linear algebra. We're starting with Module 3, "Systems of Linear Equations," because they provide a practical and intuitive entry point into the world of matrices and vectors.

In this lesson, we will focus on translating systems of linear equations into a compact and powerful form: the matrix equation $A x = b$ . This skill is fundamental. It not only simplifies the way we write complex systems but also opens the door to solving them using the elegant and efficient tools of matrix algebra, which we will explore throughout this course. By the end of this lesson, you'll be able to look at a standard system of equations and confidently represent it in matrix form.

From Systems to Vectors

Let's start with something familiar: a system of linear equations. Consider this simple example with two equations and two variables, $x_{1}$ and $x_{2}$ :

2 x_{1} + 3 x_{2} x_{1} - 4 x_{2} = 7 = - 2

Our goal is to rewrite this in a more structured way. Notice how the variables $x_{1}$ and $x_{2}$ act as "placeholders" for coefficients. We can separate the coefficients associated with each variable into column vectors:

x_{1} (21) + x_{2} (3 - 4) = (7 - 2)

This is called a vector equation. It expresses the same relationship as our original system but frames it differently. We are now looking for a linear combination of the vectors $(21)$ and $(3 - 4)$ that produces the vector $(7 - 2)$ . The "weights" of this combination are the unknown values $x_{1}$ and $x_{2}$ .

This idea of representing a system of equations in different but equivalent ways is central to linear algebra. To see a clear overview of these connections, let's watch a segment from a helpful video.

Linear Algebra 1.4.1 The Matrix Equation Ax=b

In this video, "Linear Algebra 1.4.1 The Matrix Equation Ax=b," Kimberly Brehm provides an excellent visual summary of the four equivalent ways to represent a single system of equations.

Please watch the segment from "four representations". As you watch, pay close attention to how the coefficients and constants from the system of equations are rearranged to form the augmented matrix, the vector equation, and finally, the matrix equation.

The video shows four equivalent forms. We've just covered the transition from a system of equations to a vector equation. Next, we'll make the final jump to the matrix equation, $A x = b$ .

The Matrix-Vector Product

The vector equation format, $x_{1} a_{1} + x_{2} a_{2} + \dots + x_{n} a_{n} = b$ , is so fundamental that linear algebra gives it a special, even more compact notation: the matrix-vector product.

The core idea is to bundle the column vectors $a_{1}, a_{2}, \dots, a_{n}$ into a single object called a matrix, which we'll label $A$ . We also bundle the scalar weights $x_{1}, x_{2}, \dots, x_{n}$ into a single column vector, $x$ .

The product of the matrix $A$ and the vector $x$ , written as $A x$ , is defined as the linear combination of the columns of $A$ using the corresponding entries of $x$ as weights.

Math 2331 – Linear Algebra - 1.4 The Matrix Equation Ax=b

This document from the University of Houston provides the formal definition of the matrix-vector product.

On page 3, please read the section titled "Matrix-Vector Multiplication". This definition is the cornerstone of our current topic. Notice that A\mathbf{x} is defined precisely as the linear combination we saw in our vector equation.

So, if we have:

A matrix $A$ whose columns are $a_{1}, a_{2}, \dots, a_{n}$ .
A vector $x$ with entries $x_{1}, x_{2}, \dots, x_{n}$ .

Then:

A = ∣ a_{1} ∣ ∣ a_{2} ∣ \dots ∣ a_{n} ∣, x = x_{1} x_{2} ⋮ x_{n}

And the product $A x$ is:

A x = x_{1} a_{1} + x_{2} a_{2} + \dots + x_{n} a_{n}

This might seem abstract, so let's see it developed from the ground up.

Matrix-Vector Multiplication and the equation Ax=b

In his video "Matrix-Vector Multiplication and the equation Ax=b," Dr. Trefor Bazett gives a wonderful intuitive explanation of how the notation A\mathbf{x} = \mathbf{b} naturally arises from the concept of a linear combination.

Watch the portion of the video from the beginning of his explanation up to the final definition. He starts with the familiar linear combination, breaks it down, and then masterfully reassembles it into the matrix-vector product, showing how A and \mathbf{x} are constructed.

Formulating the Equation $A x = b$

Now we can put everything together. We have established a chain of equivalence:

System of Linear Equations $⟺$ Vector Equation $⟺$ Matrix Equation $A x = b$

This equivalence is formally stated in a theorem you can find in many linear algebra texts.

1.4 The Matrix Equation Ax = b

This document from UC Berkeley concisely states the equivalence theorem.

Please read Theorem 3. It formalizes the connection we've been building, stating that the matrix equation, the vector equation, and the system of linear equations all have the same solution set.

Let's apply this to a concrete example.

Problem: Formulate the following system of linear equations as a matrix equation $A x = b$ .

8 x_{1} - x_{2} 5 x_{1} + 4 x_{2} x_{1} - 3 x_{2} = 4 = 1 = 2

Step 1: Identify the coefficients and constants.
The coefficients of $x_{1}$ form our first column vector, $a_{1}$ . The coefficients of $x_{2}$ form our second, $a_{2}$ . The constants on the right-hand side form the vector $b$ .

Step 2: Write the vector equation.
This is an intermediate step that clarifies the structure.

x_{1} 851 + x_{2} - 1 4 - 3 = 412

Step 3: Construct the matrix $A$ , the vector $x$ , and the vector $b$ .

The matrix $A$ is formed by joining the column vectors $a_{1}$ and $a_{2}$ .
The vector $x$ contains the variables.
The vector $b$ contains the constants.

A = 851 - 1 4 - 3, x = (x_{1} x_{2}), b = 412

Step 4: Write the final matrix equation.
By combining these parts, we get the final form: $A x = b$ .

851 - 1 4 - 3 (x_{1} x_{2}) = 412

This single, compact equation contains all the information of the original, more cumbersome system.

Conclusion

In this lesson, we established a crucial link between three different representations of the same problem. Let's summarize the key takeaways:

A system of linear equations can be rewritten as a vector equation, where the solution involves finding the right weights for a linear combination of column vectors.
The matrix-vector product, $A x$ , is defined as the linear combination of the columns of matrix $A$ with the entries of vector $x$ as weights.
This definition allows us to express any system of linear equations in the concise and powerful form $A x = b$ , where $A$ is the coefficient matrix, $x$ is the vector of variables, and $b$ is the vector of constants.

This formulation is more than just a notational convenience. It allows us to re-imagine a system of equations as a single entity, which we can then manipulate using the rules of matrix algebra.

In our next lesson, we will build directly on this foundation. We will introduce the augmented matrix, a structure that combines $A$ and $b$ into a single matrix. This will be our primary tool for developing a systematic algorithm, known as Gaussian elimination, to solve these systems.

Recommended resources

From Systems to Vectors

The Matrix-Vector Product

Formulating the Equation Ax=b

Conclusion

Formulating the Equation $A x = b$