Change of Basis - The Matrix of A Linear Transformation

The Matrix of A Linear Transformation

Now suppose T : V → W is a linear transformation, {α₁, ..., α_n} is a basis for V and {β₁, ..., β_m} is a basis for W. Let φ and ψ be the coordinate isomorphisms for V and W, respectively, relative to the given bases. Then the map T₁ = ψ-1 o T o φ is a linear transformation from Rn to Rm, and therefore has a matrix t; its j-th column is ψ-1(T(α_j)) for j = 1, ..., n. This matrix is called the matrix of T with respect to the ordered bases {α₁, ..., α_n} and {β₁, ..., β_m}. If η = T(ξ) and y and x are the coordinate tuples of η and ξ, then y = ψ-1(T(φ(x))) = tx. Conversely, if ξ is in V and x = φ-1(ξ) is the coordinate tuple of ξ with respect to {α₁, ..., α_n}, and we set y = tx and η = ψ(y), then η = ψ(T₁(x)) = T(ξ). That is, if ξ is in V and η is in W and x and y are their coordinate tuples, then y = tx if and only if η = T(ξ).

Theorem Suppose U, V and W are vector spaces of finite dimension and an ordered basis is chosen for each. If T : U → V and S : V → W are linear transformations with matrices s and t, then the matrix of the linear transformation S o T : U → W (with respect to the given bases) is st.