Matrices

A matrix is a rectangular grid of numbers. That sounds dull, but a matrix is really two things at once: a tidy way to store numbers, and a machine that takes in a vector and puts out another. The second view is what makes linear algebra powerful.

A grid of numbers

We write a matrix as rows and columns, for example

A = (\begin{matrix} 2 & 0 \\ 1 & 3 \end{matrix}) .

Its size is given as rows by columns; this one is $2 \times 2$ . The plainest use is bookkeeping: a spreadsheet, a table of coefficients, the pixels of an image.

The matrix-vector product

The magic is how a matrix acts on a vector. To multiply $A$ by a vector, you combine the vector's entries with the matrix's columns. For the matrix above acting on $(x, y)$ :

A (\begin{matrix} x \\ y \end{matrix}) = x (\begin{matrix} 2 \\ 1 \end{matrix}) + y (\begin{matrix} 0 \\ 3 \end{matrix}) .

Read that carefully: the output is a linear combination of the matrix's columns, weighted by the vector's entries. A matrix feeds a vector in and returns a new vector: it is a function.

Systems of equations

This is also why matrices tame systems of equations. A tangle of linear equations like

2 x + 0 y = 4, x + 3 y = 5

collapses into the single compact statement $A x = b$ , where $b = (4, 5)$ . Solving the system means finding the input vector $x$ that the machine $A$ sends to $b$ . All the messy algebra of substitution becomes one clean question about a matrix.

Combining machines

Because matrices are functions, you can chain them: apply one, then another. That chaining is matrix multiplication, and the order matters: doing $A$ then $B$ is generally not the same as $B$ then $A$ , just as putting on socks then shoes is not the same as shoes then socks. The next lesson makes the "machine" view precise: every matrix is a transformation of space.