Systems of linear equations — matrix form Ax = b: Foundations

Before you can trust the parent note, you must be able to read it. This page assumes nothing — not even what a "vector" is. We build every symbol, in order, each one leaning on the one before.

0. A number, and a name for a number

We start below the floor. A scalar is just an ordinary number: $5$ , $- 2$ , $\frac{1}{3}$ . We call it "scalar" because its only job is to scale — to stretch or shrink other things. That's the whole meaning; hold onto the word "scale".

1. Subscripts — house numbers on a street

When we have many numbers of the same kind, we don't invent 20 new letters. We reuse one letter and give each copy an address written below it. That address is a subscript.

$x_{1}, x_{2}, x_{3}$ — three different unknown numbers, all "of the $x$ family".
$a_{ij}$ — two addresses: $i$ is the row, $j$ is the column. Read it "a-eye-jay". So $a_{23}$ lives in row 2, column 3.

2. A vector — a list of numbers with a direction

Stack several scalars into a column and you get a vector.

x = [x_{1} x_{2}]

Two ways to picture it, both true and both needed later:

A list — a "to-do" of numbers ("2 scoops flour, 1 scoop sugar").
An arrow — starting at the origin $(0, 0)$ , pointing to the coordinate $(x_{1}, x_{2})$ . Its length and direction are the picture.

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

2a. Scaling and adding vectors (the two moves)

Only two operations are ever done to vectors, and every later idea is built from them.

Scale: multiply a vector by a scalar $c$ — stretch the arrow by factor $c$ (flip it if $c < 0$ ).

2 [13] = [26]

Add: put arrows tip-to-tail; the sum is the arrow from the very start to the very end.

[10] + [01] = [11]

3. The dot product — one number measuring "matched-up sum"

Take two vectors of the same length. Multiply them entry-by-entry, then add up all the products. That single output number is the dot product.

[a_{1} a_{2}] \cdot [x_{1} x_{2}] = a_{1} x_{1} + a_{2} x_{2} .

Why this exact recipe? Look at one equation from a system:

a_{1} x_{1} + a_{2} x_{2} = b_{1} .

The left side is already a dot product of the coefficient list $(a_{1}, a_{2})$ with the unknown list $(x_{1}, x_{2})$ . So the dot product is not an extra invention — it is the machine that turns a coefficient row plus the unknowns into the left-hand side of an equation. That is precisely why the parent note says "the $i$ -th entry of $A x$ is the dot product of row $i$ with $x$ ."

4. $\sum$ — a compact "add up all of these"

The symbol $== \sum ==$ (Greek capital sigma) means sum. It has three parts:

j = 1 \sum n a_{ij} x_{j} = a_{i 1} x_{1} + a_{i 2} x_{2} + \dots + a_{in} x_{n} .

below ( $j = 1$ ): the counter starts here,
above ( $n$ ): the counter stops here,
body ( $a_{ij} x_{j}$ ): plug each $j$ in and add the results.

5. The matrix — a whole grid of numbers

Stack the coefficient rows of all the equations into a rectangle: a matrix.

A = [a_{11} a_{21} a_{12} a_{22}]

Its size is $m \times n$ : $m$ rows (one per equation), $n$ columns (one per unknown). We write $A \in R^{m \times n}$ .

Two ways to slice it, both essential:

By rows — each row is a coefficient list for one equation (feeds the row view).
By columns — each column $a_{(j)}$ is a vector (feeds the column view).

6. Matrix times vector — the star of the show

Now assemble the pieces. To compute $A x$ : take each row of $A$ , dot it with $x$ (§3), and stack the results into a new vector.

[a_{11} a_{21} a_{12} a_{22}] [x_{1} x_{2}] = [a_{11} x_{1} + a_{12} x_{2} a_{21} x_{1} + a_{22} x_{2}]

So the $i$ -th entry is exactly $(A x)_{i} = j = 1 \sum n a_{ij} x_{j}$ — the dot product of row $i$ with $x$ . See how §3, §4, §5 all converge here.

Second picture — the column view. The same product regroups into a linear combination of the columns:

A x = x_{1} [a_{11} a_{21}] + x_{2} [a_{12} a_{22}] = x_{1} a_{(1)} + x_{2} a_{(2)} .

Read it as: "use $x_{1}$ scoops of column 1, plus $x_{2}$ scoops of column 2." That is scaling (§2a) then adding (§2a) — nothing new.

7. The equals sign, done properly

$A x = b$ means the vector $A x$ and the vector $b$ are the same list — entry 1 equals entry 1, entry 2 equals entry 2, and so on. One vector equation secretly holds $m$ scalar equations at once. That is the "bookkeeping" the parent note celebrates.

8. Words you'll meet in the answer

These name the outcomes; you don't need to master them yet — the linked notes do that — but you must recognise the words.

Definition The vocabulary of solving

Span — every vector you can reach by scaling-and-adding a given set. The reachable region. → Column space and null space
Column space $Col (A)$ — the span of $A$ 's columns; the set of all possible $A x$ outputs. $b$ must live here for a solution to exist. → Column space and null space
Null space — all inputs $x$ with $A x = 0$ ; these are the "free wiggle" directions. → Column space and null space
Rank — the count of genuinely independent rows/columns. It decides existence and uniqueness. → Rank of a matrix
Linear independence — no vector in a set is a scale-and-add mix of the others. → Linear independence
Determinant $det A$ — one number (square matrices only) that is nonzero exactly when $A$ is invertible. → Determinant
Inverse $A^{- 1}$ — the "undo" matrix giving $x = A^{- 1} b$ . → Inverse of a matrix

The prerequisite map

Each arrow means "you need the left box before the right box makes sense." Notice everything funnels into $A x$ , then into $A x = b$ .

Equipment checklist

Recall Self-test: can you say each in one breath?

What is a scalar? ::: A single real number whose job is to scale things. What do the two subscripts in $a_{ij}$ mean? ::: Row $i$ , column $j$ — an address, not the number "ij". What are the only two operations on vectors? ::: Scaling (stretch/flip by a scalar) and adding (tip-to-tail). How do you compute a dot product? ::: Multiply entries in matching positions, then sum them into one scalar. What does $\sum_{j = 1}^{n} a_{ij} x_{j}$ spell out? ::: $a_{i 1} x_{1} + a_{i 2} x_{2} + \dots + a_{in} x_{n}$ . What are the two ways to read $A x$ ? ::: Rows dotted with $x$ , OR a weighted sum of the columns of $A$ . What does $A x = b$ really assert? ::: The output vector $A x$ matches $b$ entry-by-entry — $m$ equations in one line. Why is $x = A^{- 1} b$ not "dividing by $A$ "? ::: Matrices have no division; $A^{- 1}$ is the separate undo-matrix that exists only when $A$ is square with $det A \neq = 0$ .

Ready? Then return to the parent topic and the three views will read like plain English.