Linear transformations

A map is linear if applying it to any linear combination gives the same result as applying it separately and combining the outputs. Consequently a linear map sends zero to zero. Multiplication by a fixed matrix is linear, including rectangular matrices. The familiar formula $f(x)=2x+3$ is affine rather than linear in this sense: its offset means that zero does not remain zero.

Once you know the images of all basis vectors, you know the entire linear map. Write an input as a combination of basis vectors, distribute the map through that combination, and collect the outputs. This argument explains the column convention: in standard coordinates, column $j$ of a matrix is where the $j$th input basis vector lands. Matrix multiplication is the bookkeeping of this geometric statement.

The nullspace consists of all inputs mapped to zero. If a nonzero vector lies there, two distinct inputs separated by that vector have exactly the same output. The transformation has discarded information. The column space, also called its image, contains every output that can be reached. Its dimension is the rank. A measurement system can lose distinctions in its input while still covering every possible output channel.

Rank and nullity account for the input dimension together. An independent input direction either contributes to an observable output or belongs to the invisible combinations described by the kernel. For a matrix with more columns than rows, a nontrivial nullspace is unavoidable. For a square matrix, zero nullity is equivalent to invertibility; a rectangular map needs more careful distinctions between left inverses and right inverses.

If $B$ acts first and $A$ second, the composed matrix is $AB$. Dimensions must match, and reversing the order generally changes the transformation or makes the product undefined. Stretching a picture horizontally and then rotating it does not usually produce the same picture as rotating first. This is a concrete reason that multiplication of matrices differs from multiplication of ordinary numbers.

Changing bases changes a matrix representation but not the abstract map. For one endomorphism, if columns of an invertible matrix $P$ are the new basis vectors, the new representation is $P^{-1}AP$. This coordinate flexibility later helps expose eigenmodes and quantum operators. A learned neural-network layer with a bias instead has an affine component; calling the whole layer linear would conceal that distinction.