Invertibility of linear transformations

The fact that linear transformations can be represented as matrices, allows us to interpret many properties of linear transformations as analogous to properties of their standard matrices. Being able to represent linear transformations by matrices allows us to extend theorems about matrices to theorems about linear transformations, as long as the underlying vector spaces are finite-dimensional.
For example, we say that a linear transformation T : V → W is invertible if there exists a linear transformation T−1 : T: W → V such that

T−1 (T (v)) = v for all v ∈ V and
T (T−1 (w)) = w for all w ∈ W.

In other words, under composition of functions T−1 ◦ T = IV and T ◦ T−1 = IW. If T : V → W is an invertible linear transformation, T is called an isomorphism, and we say V and W are isomorphic. Our intuitive understanding of the inverse of a linear transformation is identical to that of matrices (T−1 is the linear transformation that “undoes” what T “does”), and the following theorem says that invertibility of T can in fact be determined from its standard matrix:

Theorem 3.4.1.

We show in Exercise 3.4.2 that if dim(V) ≠ dim(W) then T cannot possibly be invertible.

Exercise 1.2.30. Let B be a basis of an n-dimensional vector space V and let T : V → V be a linear transformation. Show that [T ] B = I n if and only if T = IV.

Exercise 3.4.3. Let V be a finite-dimensional vector space over a field 𝔽 and suppose that B is a basis of V.

  1. Show that [v + w]B = [v]B + [w]B for all v, w ∈ V.

  2. Show that [cv]B = c [v]B for all v ∈ V and c ∈ 𝔽.

  3. Suppose v, w ∈ V. Show that [v]B = [w]B if and only if v = w.

This means that the function T : V → 𝔽n defined by T (v) = [v]B is an invertible linear transformation.

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