What do the coordinates of a vector actually mean?

They are counts. The coordinates (3, 2) mean three of the first basis vector plus two of the second. Coordinates are therefore a statement about the chosen basis, not a property of the vector itself: change the basis and the same movement gets different numbers.

Why are the columns of a matrix where the basis vectors land?

Because multiplying a matrix by the first basis vector (1, 0) selects its first column, by the second basis vector (0, 1) selects its second, and so on. A matrix is therefore nothing but a list of the new positions of the basis vectors, written side by side.

Essay · Linear algebra

Coordinates are counts of building blocks

Q: What is a basis vector?

A basis vector is one of the building blocks chosen to describe every other vector in a space. In the plane the standard choice is one step east and one step north. A set of vectors is a basis when it can reach every point in the space (it spans) and contains no redundant member (it is linearly independent).

Q: What makes a transformation linear?

Grid lines stay straight, parallel and evenly spaced, and the origin stays put. The consequence is that a transformation is completely determined by where the basis vectors land: every other point is built from them, and building is preserved.

what a basis vector is, and why a matrix is a list of landing spots

Updated September 2026

A vector is a movement, and a basis is the set of building blocks you're allowed to use to describe it. A vector's coordinates are counts: (3, 2) means three of the first block plus two of the second. Because every point is built this way, knowing where the basis vectors land tells you where everything lands — which is exactly what a matrix records.

Most of us were handed coordinates first and vectors second: a point is a pair of numbers, and a vector is an arrow you draw between two of them. It works well enough to pass an exam and badly enough that matrices stay mysterious. Turning the order around fixes it.

A vector is a movement, not an arrow

Start with the thing being described. "Three steps east, two steps north" is a movement. It doesn't care where you started, and it isn't a location — it's an instruction. The arrow that gets drawn is a picture of that instruction, and the pair of numbers is a written record of it. Both are descriptions; the movement is the thing.

Once a vector is a movement, the two basic operations stop needing rules. Adding two vectors is performing one movement and then the other. Scaling a vector by 3 is doing the same movement three times. Nothing about arrows had to be memorised.

So what is a basis vector?

To describe any movement with numbers you first have to agree on the blocks you're building from. In the plane the usual agreement is two: one step east and one step north, written (1, 0) and (0, 1) and often called i and j. Those are the basis vectors.

A set of blocks earns the name basis when it passes two tests:

It spans the space — some combination of the blocks reaches every point there is.
It is linearly independent — no block is redundant; you can't build any one of them out of the others.

Two non-parallel arrows in the plane pass both tests. Two parallel ones fail the first: everything they build lies along a single line, so most of the plane is unreachable. Three arrows in the plane fail the second: one of them was always expressible with the other two.

Which makes coordinates almost embarrassingly simple

A linear combination is just "use some of this block and some of that one": 3i + 2j. And the coordinates (3, 2) are the receipt — how many of each block went in. That's all a coordinate has ever been.

Which also explains something that confuses people later: coordinates are a statement about the basis, not about the vector. Pick different blocks and the same physical movement gets a different pair of numbers. Nothing moved; the units of description changed. It's the same trick as calling one quantity 1 and measuring the other against it.

What makes a transformation "linear"?

A transformation moves every point of the space at once. It counts as linear when the grid survives the move: grid lines stay straight, stay parallel, stay evenly spaced, and the origin doesn't budge.

That restriction is what buys you everything else. If a point was built as 3i + 2j before the move, it is built the same way out of the new i and j afterwards. Building is preserved. So you don't have to track every point — track the blocks, and the rest follows.

Why a matrix's columns are the landing spots

Now the notation stops being arbitrary. A matrix is a list of where the basis vectors ended up, written side by side as columns. Multiply a 2×2 matrix by (1, 0) and the arithmetic hands back its first column; multiply by (0, 1) and you get the second. The matrix was never anything but that list.

Each matrix reads as "`i` goes here, `j` goes there". Written out in rows, as above, those landing spots sit down the *columns* — first column where `i` went, second where `j` went.
Transformation	Where `i` lands	Where `j` lands	Matrix, written in rows
Do nothing (identity)	(1, 0)	(0, 1)	`[[1, 0], [0, 1]]`
Stretch everything by 2	(2, 0)	(0, 2)	`[[2, 0], [0, 2]]`
Rotate 90° counter-clockwise	(0, 1)	(−1, 0)	`[[0, −1], [1, 0]]`
Shear to the right	(1, 0)	(1, 1)	`[[1, 1], [0, 1]]`
Reflect across the x-axis	(1, 0)	(0, −1)	`[[1, 0], [0, −1]]`
Flatten onto the x-axis	(1, 0)	(0, 0)	`[[1, 0], [0, 0]]`

The last row is worth a second look. Once j lands on the origin, one of the blocks has been destroyed, and everything in the plane collapses onto a line. There's no way back, because countless different points now share a single landing spot. That is what a determinant of zero, a non-invertible matrix, and a "loss of a dimension" all describe — one sentence, three pieces of jargon.

Why a calculus series stops to do this

Because the next step needs it. With one input, nudging it is unambiguous — there's only forward. On a surface with two inputs, the moment you nudge a point you have to say which way, and a direction with a size is a vector. The directional derivative is the nudge ratio with that vocabulary installed.

Go stand on it

Coordinates are easier to trust when they're moving under your feet. Open World Discover keeps a live (x, y, z) readout on screen while you walk a function's graph, and has an origin axis toggle — east red, north green, up blue — so you can see the blocks themselves.