Linear Algebra · step 7 of 13

Linear maps & their matrices

A map T is linear when T(u + v) = T(u) + T(v) and T(cv) = cT(v). Every linear map between finite-dimensional spaces is a matrix in disguise: put the images of the basis vectors in as COLUMNS and you have it. Composition of maps then corresponds exactly to matrix multiplication, which is where that definition of multiplication comes from.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Check linearity from the two axiomsOr the single condition T(au + bv) = aT(u) + bT(v). Note T(0) = 0 is necessary, so any map with a constant term fails.
Compute the images of the basis vectorsT(e₁), T(e₂), … Each becomes a column of the matrix, in order.
Assemble the matrixThe matrix of T with respect to the standard bases is [T(e₁) | T(e₂) | … ].
Compose by multiplyingThe matrix of S∘T is (matrix of S)(matrix of T), in that order.

Worked example

Find the matrix of the map ℝ² → ℝ² rotating vectors 90° anticlockwise.

Rotation is linear: it preserves sums and scalar multiples.
e₁ = (1,0) rotates to (0,1).
e₂ = (0,1) rotates to (−1,0).
Place these as columns.

Answer. [[0, −1], [1, 0]]. Check on (1,1): the product is (−1,1), which is (1,1) rotated a quarter turn ✓.

Where marks get dropped

These are the specific errors that cost credit on linear maps & their matrices questions — QED's rubric penalises each of them separately.

Writing the images as rows instead of columns, which gives the transpose — a different map.
Assuming a map with a constant offset is linear. T(x) = x + 1 fails T(0) = 0; it is affine, not linear.
Composing in the wrong order. The matrix of S∘T applies T first, so T’s matrix is on the right.

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Linear maps & their matrices — frequently asked questions

Why are the images of basis vectors enough?

Because every vector is a unique linear combination of the basis, and linearity carries that combination through. Knowing T on a basis determines T everywhere.

Is every matrix a linear map?

Yes — v ↦ Av is linear for any matrix A, and every linear map between finite-dimensional spaces arises this way once bases are fixed.

What if the bases are not standard?

The matrix changes. Express each T(basis vector) in terms of the target basis, and those coordinate vectors are the columns.

The rest of Linear Algebra

Matrices, systems, determinants, eigenvalues. Each subtopic below has its own method, worked example and mark-losing traps.

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