Systems of linear equations — matrix form Ax = b: Visual walkthrough

Step 1 — What is a "linear equation", really?

WHAT. Start with the simplest possible balance condition on two unknown numbers, call them $x$ and $y$ :

how much x counts a x + how much y counts b y = the target total c .

Here $a$ , $b$ , $c$ are known fixed numbers; $x$ , $y$ are the unknowns we hunt for. "Linear" means the unknowns appear plain — never squared, never multiplied together, never inside anything curvy.

WHY start here. Before we pack anything into a matrix, we must know what ONE equation looks like as a picture. Everything later is just stacking copies of this.

PICTURE. All the pairs $(x, y)$ that satisfy one such equation form a straight line. That is the whole reason for the word linear.

Figure — Systems of linear equations — matrix form Ax = b

Step 2 — Two equations = two lines = the "row view"

WHAT. Now demand two balance conditions at once:

a_{11} x + a_{12} y a_{21} x + a_{22} y = b_{1} (line 1) = b_{2} (line 2)

Read the subscripts like a seat number: $a_{21}$ means row 2, column 1 — "the coefficient of the 1st unknown in the 2nd equation". The letter $b_{1}$ is the target of equation 1.

WHY. A single equation has a whole line of answers — too many. Adding a second equation is how we pin things down: a point must sit on both lines simultaneously.

PICTURE. Two lines in the plane. The solution is their single crossing point. This is the row view: each row of the system is a line (in 3D, a flat plane; the fancy word is hyperplane), and solving = intersecting them.

Step 3 — Pack the numbers: birth of $A$ , $x$ , $b$

WHAT. We now do pure bookkeeping — no new maths, just tidier storage. Gather the four coefficients into a grid, the two unknowns into a stack, the two targets into a stack:

A = coefficient grid [a_{11} a_{21} a_{12} a_{22}], x = unknowns [x y], b = targets [b_{1} b_{2}] .

A matrix is just a rectangle of numbers with named slots; a vector here is a single column of numbers. We define the product $A x$ so that stacking it and setting it equal to $b$ reproduces our two equations exactly:

A x = [a_{11} x + a_{12} y a_{21} x + a_{22} y] =! [b_{1} b_{2}] = b .

The little $=!$ means "we require this to hold". Row $i$ of $A x$ is the dot product of row $i$ of $A$ with $x$ — multiply matching entries, add them up. See Matrix multiplication.

WHY define it this way. Because we want the compressed symbol $A x = b$ to mean the same thing as the two long equations. The definition of the product is reverse-engineered to make that true — nothing more mysterious than that.

PICTURE. The grid, the two stacks, and an arrow showing how row 1 "reaches across" $x$ to make equation 1.

Step 4 — Re-slice $A x$ by COLUMNS: mixing arrows

WHAT. Same product $A x$ , cut a different way. Instead of reading across rows, collect the terms that carry $x$ and the terms that carry $y$ :

A x = [a_{11} x + a_{12} y a_{21} x + a_{22} y] = x column 1 [a_{11} a_{21}] + y column 2 [a_{12} a_{22}] .

Call the columns $a_{(1)} = [a_{11} a_{21}]$ and $a_{(2)} = [a_{12} a_{22}]$ . Then

A x = x a_{(1)} + y a_{(2)} .

The unknowns $x$ and $y$ became scoop counts — how many of arrow $a_{(1)}$ , how many of arrow $a_{(2)}$ .

WHY this re-slice matters. It converts algebra into geometry: $A x$ is a linear combination of the column arrows. Solving $A x = b$ becomes a visual question — "how many scoops of each column arrow do I add to land exactly on the target arrow $b$ ?" This is the column view, and it is the 80/20 of the whole chapter.

PICTURE. Two column arrows starting at the origin; stretch/shrink each, then add tip-to-tail. The scoop counts $(x, y)$ that make the sum reach $b$ are the solution.

Step 5 — The existence rule: is $b$ reachable?

WHAT. $A x = b$ has a solution exactly when the target $b$ lies somewhere in that reachable set:

A x = b solvable ⟺ b \in Col (A) .

WHY. If no combination of scoops of the columns ever points at $b$ , then no $x$ solves the system — full stop. The columns simply cannot manufacture that arrow.

PICTURE. When the two columns point in different directions they fill the whole plane, so every $b$ is reachable (left). When they are parallel, they only ever reach along a single line — a $b$ off that line is unreachable (right). "Different directions" is the visual meaning of Linear independence.

Step 6 — The transformation view: which input lands on $b$ ?

WHAT. Read $A$ as a machine $x \mapsto A x$ : feed in an arrow, get out an arrow. Solving $A x = b$ asks the machine's question in reverse: which input arrow does the machine send to $b$ ?

WHY. This lens explains uniqueness. If the machine squashes the plane onto a line (parallel columns), many different inputs get flattened onto the same output — so a reachable $b$ has many preimages. If the machine keeps the plane full and 2D, each output has exactly one input.

PICTURE. A grid of input arrows on the left; the machine warps them into output arrows on the right. Highlight the input that maps to $b$ .

Step 7 — The three verdicts (all cases covered)

WHAT. Combine everything. Let $r = rank (A)$ = number of genuinely independent directions the columns give (see Rank of a matrix), $n$ = number of unknowns, and compare with the augmented grid $[A ∣ b]$ (glue $b$ on as an extra column):

Picture	Condition	Verdict
Lines cross at one point	$\operatorname{rank}A=\operatorname{rank}[A,	,b]=n$
Same line twice	$\operatorname{rank}A=\operatorname{rank}[A,	,b]<n$
Parallel, never meet	$\operatorname{rank}A<\operatorname{rank}[A,	,b]$

WHY. Gluing $b$ and finding the rank jumps means $b$ demanded a fresh direction the columns can't supply → inconsistent. If ranks match but there are fewer independent equations than unknowns, leftover unknowns float free ( $n - r$ of them).

PICTURE. The three canonical row pictures side by side.

The one-picture summary

Below, the SAME product $A x$ is read all three ways — rows crossing, columns mixing, machine mapping — all agreeing on one answer. That agreement is linear algebra.

Recall Feynman retelling — say it to a friend

You have two dials, $x$ and $y$ . Each dial controls how many scoops of one ingredient arrow (a column of $A$ ) you pour into a bowl. Adding the poured arrows tip-to-tail gives one final arrow. The recipe asks: set the dials so the final arrow lands exactly on the target $b$ . If the two ingredient arrows point in different directions, they can build anything in the plane, so there is always exactly one dial setting — unique. If both ingredients point the same way, you can only ever build arrows along that one line: a target off the line is impossible (no solution), while a target on the line can be built in endlessly many ways (turn one dial up and the other down to compensate). Reading the very same numbers as crossing lines (row view) or as a squashing machine (transformation view) tells the identical story — that's why $A x = b$ is the whole subject in one line.

Connections

Matrix multiplication — the row-dot-column rule that defines $A x$ (Step 3).
Column space and null space — the reachable set and the floating directions (Steps 4–5).
Rank of a matrix — counts independent directions; decides the verdict (Step 7).
Linear independence — "columns point different ways" made precise (Step 5).
Gaussian elimination — the machine that computes the ranks and scoops.
Determinant — square-case test for "columns fill the plane".
Inverse of a matrix — the unique-case formula $x = A^{- 1} b$ .