Linear Transformations and Matrices

References are to Anton–Rorres, 7th Edition See also “Coordinate Vectors”, in this series

We wish to describe a general linear transformation T: V → W in terms of coordinate systems on V and W. Suppose we are given bases B = {b_{1}, b_{2}, …, b_{n}} of V and C = {c_{1}, c_{2}, …, c_{m}} of W. We use the resulting linear isomorphisms L_{B}: R^{n} → V and L_{C}: R^{m} → W to convert T to a linear transformation R^{n} → R^{m}, which we know how to handle. Diagramatically, we have

[figure: V; W; R^{n}; R^{m}; T; L_{B}; L_{C}]

Definition 1 The matrix of T: V → W with respect to the bases B and C is the matrix A of the composite linear transformation L_{C}^{−1} ^{∘} T ^{∘} L_{B}: R^{n} → R^{m}. We write it [T]_{C,B} (the order here may seem perverse; but see below). If W = V and C = B, we write simply [T]_{B}.

Remark If W = V but we choose different bases B and C of V, the matrix [I]_{B,C} of the identity linear transformation I: V → V is the transition matrix P from C to B. This is Theorem 8.5.1. To see this, just compare the definitions (and watch the order of B and C!).

To make A explicit, we need to compute the effect of this linear transformation on e_{i}. First, L_{B}(e_{i}) = b_{i}. Then we apply T, to get T (b_{i}). Finally, L_{C}^{−1}(T (b_{i})) is by definition the coordinate vector [T (b_{i})]_{C} of T (b_{i}) with respect to C. Thus the matrix A of T is

A = [[T (b_{1})]_{C} |[T (b_{2})]_{C} | … |[T (b_{n})]_{C}]  (2)

It does what we expect on coordinate vectors.

Theorem 3 If A is the matrix of T: V → W with respect to bases B and C, then

A[v]_{B} = [T (v)]_{C}  for all v ∈ V.  (4)

Proof This is simply the translation of the statement

(L_{C}^{−1} ^{∘} T ^{∘} L_{B})(L_{B}^{−1}(v)) = L_{C}^{−1}(T (v)).

Composition Matrices behave correctly under composition of linear transformations.

Theorem 5 (=Thm. 8.4.2) Let V, W, and X be vector spaces with bases B, C and D respectively. If A is the matrix of the linear transformation T: V → W with respect to B and C, and A′ is the matrix of the linear transformation U: W → X, then the matrix of the composite linear transformation U ^{∘} T: V → X is the product matrix A′A. In symbols, [U ^{∘} T]_{D,B} = [U]_{D,C} ^{∘}[T]_{C,B}.

Proof This is the translation of the statement

L_{D}^{−1} ^{∘}(U ^{∘} T)^{∘} L_{B} = (L_{D}^{−1} ^{∘} U ^{∘} L_{C})^{∘}(L_{C}^{−1} ^{∘} T ^{∘} L_{B}).

The relevant diagram is

[figure: V; W; X; R^{n}; R^{m}; R^{p}; T; U; L_{B}; L_{C}; L_{D}]

Similarity We need to know the effect of a change of basis on the matrix of a linear operator T: V → V.

Theorem 6 (=Thm. 8.5.2) Let A be the matrix of the linear operator T: V → V with respect to the basis B of V. Let C be another basis of V, with P the transition matrix from C to B. Then the matrix of T with respect to C is P^{−1}AP. In symbols,

[T]_{C} = P^{−1}[T]_{B}P.  (7)

Proof This translates the obvious statement

L_{C}^{−1} ^{∘} T ^{∘} L_{C} = (L_{C}^{−1} ^{∘} L_{B})^{∘}(L_{B}^{−1} ^{∘} T ^{∘} L_{B})^{∘}(L_{B}^{−1} ^{∘} L_{C})

Change of basis The general case is no harder.

Theorem 8 Let B and B′ be bases of V, and P the transition matrix from B′ to B. Let C and C′ be bases of W, and Q the transition matrix from C′ to C. If the matrix of T: V → W with respect to B and C is A, then the matrix of T with respect to B′ and C′ is Q^{−1}AP.

Proof This translates the statement

L_{C′}^{−1} ^{∘} T ^{∘} L_{B′} = (L_{C′}^{−1} ^{∘} L_{C})^{∘}(L_{C}^{−1} ^{∘} T ^{∘} L_{B})^{∘}(L_{B}^{−1} ^{∘} L_{B′}).