Foundations — Scalar multiplication

This page is the ground floor. The parent note Scalar multiplication throws a lot of notation at you — $a_{ij}$, $m\times n$, $\mathbb{R}$, $[\cdot ]$, the zero matrix $O$. We will not use a single one of those before it is built from nothing. Read top to bottom; each block only uses ideas from the blocks above it.

1. A number (a "scalar")

Why does the topic need a special word "scalar"? Because we are about to meet grids of numbers. We need one word for "a lonely single number" ($k$) and a different word for "a whole grid" ($A$), so we never confuse them. The lonely single number is the scalar.

The picture of a scalar is a point on a number line: its distance from $0$ is its size, its side tells its sign.

You already know scalars from 2.2.04-Vectors-and-scalars — there a scalar is contrasted with a vector (something with direction). Here we contrast it with a grid.

2. Multiplying one number by another = stretching

This is the seed of the whole topic. Before scaling a grid, be crystal-clear on scaling one number.

• $k=2$: the number gets twice as far from $0$ (a stretch).
• $k=\frac{1}{2}$: half as far from $0$ (a shrink).
• $k=-1$: same distance, opposite side (a flip through $0$).
• $k=0$: lands exactly on $0$, no matter where it started.
• $k=1$: doesn't move at all.

Notice all five behaviours above — stretch, shrink, flip, collapse-to-zero, stay-put. Every one of them will happen to a whole grid later, cell by cell. Nothing new is added; it's the same move done many times.

3. A row and a column of numbers

Before a full grid, meet its two building strips.

The picture: a row is numbers standing shoulder to shoulder; a column is numbers stacked head to foot. The topic needs both words because we will count how many rows and how many columns a grid has — that count is its shape.

4. The grid: a matrix, and the bracket $[]$

Why the bracket? So the eye instantly sees one thing (a grid $A$), not four loose numbers. This is exactly the "container" the parent note talks about. Full detail lives in 2.6.01-Introduction-to-matrices.

The picture is a filing cabinet: fixed rows and columns of pigeon-holes, one number in each hole.

5. Naming a spot: $a_{ij}$

Now the trickiest piece of notation the parent leans on hardest.

Read it aloud as "$a$, row $i$, column $j$." So in $A=\left[\begin{array}{cc}2& -3\\ 5& 0\end{array}\right],$

• $a_{11}=2$ (row 1, column 1),
• $a_{12}=-3$ (row 1, column 2),
• $a_{21}=5$ (row 2, column 1),
• $a_{22}=0$ (row 2, column 2).

Why does the topic need $a_{ij}$? Because scalar multiplication happens to every hole at once, and $a_{ij}$ is the one symbol that can say "take whatever number lives in hole $(i,j)$" without writing them all out. It's a placeholder that stands for all the entries simultaneously.

6. The shape: $m\times n$

The matrix $A=\left[\begin{array}{cc}2& -3\\ 5& 0\end{array}\right]$ has 2 rows and 2 columns, so it is a $2\times 2$ matrix. The grid $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ is $2\times 3$ (2 rows, 3 columns).

The picture: $m\times n$ is the frame size of the filing cabinet. Scalar multiplication is going to change what's inside the holes but never the frame — so we need a name for the frame to say "the frame stays the same."

7. Two shorthand symbols: $[a_{ij}{]}_{m\times n}$ and $\mathbb{R}$

Why introduce $\mathbb{R}$? So the topic can say "any real scalar is allowed" in three symbols instead of a sentence. The $\in$ symbol just means "belongs to / is a member of."

8. The zero matrix $O$

Its picture: the filing cabinet with every drawer empty (holding $0$). We need it because when the scalar is $k=0$, every hole collapses to $0$ (recall the "collapse-to-zero" move in §2), so the whole grid becomes $O$. The parent note writes this as $0\cdot A=O$.

9. Putting it together — reading $kA=[k{a}_{ij}{]}_{m\times n}$

Now the parent's headline formula uses nothing you haven't built:

$kA=[k{a}_{ij}{]}_{m\times n}$

Read it slowly, symbol by symbol, using the sections above:

• $k$ — a lonely scalar, a point on the line (§1), a stretch factor (§2).
• $A$ — the grid (§4).
• $a_{ij}$ — whatever number lives in row $i$, column $j$ (§5).
• $k{a}_{ij}$ — that hole's number stretched by $k$ (§2), the single-number move done in one hole.
• $[\cdots {]}_{m\times n}$ — collect all those stretched holes back into a grid of the same frame size (§6, §7).

So the formula says, in full English: "To make $kA$, walk every hole of the grid and replace its number by that number times $k$, keeping the frame unchanged." That is the entire topic.

Prerequisite map

This foundation feeds forward: once you can scale a grid you can combine it with matrix addition, meet the harder matrix multiplication, collect the shared rules in properties of matrix operations, and later see grids act as linear transformations.

Equipment checklist

Cover the right side and answer each before moving on to the parent topic.