[Linear Algebra Series Part 9] Linear Systems and Ax = b

What this post covers

This post rewrites systems of linear equations in the form Ax = b.

Why several equations can be grouped into one matrix equation
What A, x, and b each mean
How to read Ax = b as both a transformation question and a column-combination question
Why this viewpoint prepares us for elimination and space-based ideas

Key terms

system of equations: a problem that asks several linear conditions to hold at once
matrix: the structure that stores the coefficients
vector: the grouped form of both unknowns and outputs

Core idea

At first, many people solve linear systems one equation at a time. Linear algebra asks us to step back and treat the whole system as one object.

From the previous post, remember that Ax means a linear combination of the columns of A, weighted by the entries of x. So when we write Ax = b, we are already asking a matrix-vector multiplication question.

That object is

Ax = b

Here:

A is the coefficient matrix
x is the unknown vector
b is the result vector

For example,

x + 2y = 5
3x + 4y = 11

can be written as

[1 2] [x] = [ 5]
[3 4] [y]   [11]

This changes the question. Instead of “solve two equations,” we ask:

can the matrix A produce the output b?
if so, which input vector x does it come from?
is that input unique, or are there many?

The column-combination viewpoint

From the previous post, remember that Ax is a linear combination of the columns of A.

If A = [a1 a2], then

Ax = x1 a1 + x2 a2

So Ax = b is asking whether the columns of A can be combined to make b.

That means the solvability condition is precise: Ax = b has a solution if and only if b belongs to the column space of A. For now, you can read column space in plain language as “the set of vectors you can build from the columns of A."

A note on the homogeneous case

When b = 0, we get

Ax = 0

This is called a homogeneous system. It always has the trivial solution x = 0. The null space is the set of all solutions to Ax = 0, so this special case will matter a lot later.

Step-by-step examples

Example 1) Two equations as one structure

Take

x + y = 4
2x + 3y = 9

Then

A = [1 1
     2 3]

x = [x
     y]

b = [4
     9]

Now the system becomes one question: which input vector x does the transformation A send to the output b? In this example, the answer is x = [3; 1].

Example 2) Reading by columns

If the columns of A are a1 and a2, then the system is

x1 a1 + x2 a2 = b

So we are not just solving equations. We are checking whether b can be built from the columns of A.

Example 3) A simple resource model

Suppose ingredient A contributes 2g of protein and 3g of carbohydrates, while ingredient B contributes 1g of protein and 5g of carbohydrates. If you want 7g of protein and 19g of carbohydrates, then

[2 1] [amount of A] = [ 7]
[3 5] [amount of B]   [19]

This is still Ax = b.

So the same mathematical structure appears in recipes, mixture design, and many estimation problems.

Math notes

Ax = b can be read through the lens of linear transformation and through the lens of column combinations.
The equation has a solution if and only if b lies in the column space of A.
Even if a solution exists, it may not be unique. The solution is unique exactly when the null space contains only the zero vector.
A solvable system is called consistent. A system with no solution is inconsistent.

So Ax = b is both a computation problem and a space-membership problem.

Common mistakes

Thinking the number of equations and variables tells the whole story

Not by itself. Rank, pivots, and the matrix structure matter more.

Treating `A`, `x`, and `b` as mere symbols

It is better to read A as structure, x as the unknown input, and b as the observed output.

Confusing the problem with the algorithm

Ax = b is the problem. Gaussian elimination is one method for solving or analyzing it.

Practice or extension

Rewrite 2x + y = 7, x - y = 1 as Ax = b.
Rewrite a 3-variable system as Ax = b.
Create one example of a homogeneous system Ax = 0.
Explain in words what it means to say “b is a linear combination of the columns of A.”

Wrap-up

This post introduced the system form Ax = b.

Several equations can be grouped into one matrix equation.
A is structure, x is input, and b is output.
Solving the system also means asking whether b can be built from the columns of A.
The homogeneous case Ax = 0 always has the trivial solution.

Next, we will study Gaussian elimination, the standard procedure for turning Ax = b into a form that exposes the solution structure.