4.2 Change of basis

Previously in this section, we have seen that any linear transformation can be described with the images of the basis vectors (see Section 4.1.1). This gave us the matrix representation that we use all the time. However, this very much depends on the choice of basis. Different bases yield different matrices for the same transformation.

For instance, let’s take a look at f : ℝ² →ℝ², which maps e₁ = (1,0) to the vector (2,1) and e₂ = (0,1) to (1,2). Its matrix in the standard orthonormal basis E = {e₁,e₂} is given by

Figure 4.3: The linear transformation f, defined by (5.2)

The effect of f is visualized in Figure 4.3.

What if we select a different basis, say P = {p₁ = (1,1),p₂ = (−1,1)}? With a quick calculation, we can check that

In other words, f(p₁) = 3p₁ + 0p₂ and f(p₂) = 0p₁ + p₂. This is visualized by Figure 4.4.

Figure 4.4: The effect of f on p₁ = (1,1) and p₂ = (−...