Linear Transformations

Linear functions are the simplest functions defined on vector spaces.

Definition 2.1.1. Let V and W be vector spaces over the field 𝕂. A linear transformation T: V → W is a function (map, transformation), which satisfies the following two conditions:

  1. T(v1 + v2) = T(v1) + T(v2),  for every v1, v2 ∈ V

  2. T(λv) = λT(v),   for every v ∈ V and scalar λ ∈ 𝕂.

We will denote the collection of all linear maps from V to W by ℱ(V, W). □

Example 2.1.2. The trace is the function tr : Mn(F) → F that adds up the diagonal entries of a matrix:

tr(A) := a1,1 + a2,2 + ··· + an,n   ∀A ∈ Mn (F).

Show that the trace is a linear transformation.

Solution. We need to show that the two properties of Definition 2.1.1 hold:

  1. Tr(A+B) = a1,1 + b1,1 + a2,2 + ··· + an,n + bn,n = tr(A) + tr(B);

  2. tr(cA) = ca1,1 + ··· + can,n = ctr(A).

It follows that the trace is a linear transformation. ■

Example 2.1.3. Define T : Mn,n ⟶ Mn,n by T(A) = AT. Show that T is a linear transformation.

Solution. We check that, for A and B in Mnn and scalars c,

T(A + B) = (A + B)T = AT + BT = T(A) + T(B)

and

T(cA) = (cA)T = cAT = cT(A)

Therefore, T is a linear transformation. ■

Example 2.1.4 The derivative is a linear transformation, that is, the function D: D → F defined by D(f) = f' is a linear transformation.

We need to show that the two properties of Definition 2.1.1 hold. That is, we need to show that (f + g)' = f' + g' and (cf)' = cf' for all f, g ∈ D and c ∈ ℝ. We have already proved these properties studying the algebra of derivatives, so we do not prove them again here.  ■

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