Coordinate Vectors

References are to Anton–Rorres, 7th Edition

In order to compute in a general vector space V, we usually need to install a coordinate system on V. In effect, we refer everything back to the standard vector space R^{n}, with its standard basis E = {e_{1}, e_{2}, …, e_{n}}. It is not enough to say that V looks like R^{n}; it is necessary to choose a specific linear isomorphism.

We consistently identify vectors x ∈ R^{n} with n×1 column vectors. We know all about linear transformations between the spaces R^{n}.

Theorem 1 (=Thm. 4.3.3)

(a) Every linear transformation T: R^{n} → R^{m} has the form

T (x) = Ax for all x ∈ R^{n}  (2)

for a unique m×n matrix A. Explicitly, A is given by

A = [T (e_{1})|T (e_{2})| … |T (e_{n})] .  (3)

(b) Assume m = n. Then T is an invertible linear transformation if and only if A is an invertible matrix, and if so, the matrix of T^{−1} is A^{−1}.

We call A the matrix of the linear transformation T.

Coordinate vectors The commonest way to establish an invertible linear transformation (i. e. linear isomorphism) between R^{n} and a general n-dimensional vector space V is to choose a basis B = {b_{1}, b_{2}, …, b_{n}} of V. Then we define the linear transformation L_{B}: R^{n} → V by

L_{B}(k_{1}, k_{2}, …, k_{n}) = k_{1}b_{1} + k_{2}b_{2} + … + k_{n}b_{n} in V.  (4)

So L_{B}(e_{i}) = b_{i} for each i.

The purpose of requiring B to be a basis of V is to ensure that L_{B} is invertible. To get R(L_{B}) = V, we need B to span V; and to get L_{B} to be 1–1, we need B to be linearly independent. If we already know dim(V) = n, Thm. 5.4.5 shows that it is enough to verify either of these conditions; then the other will follow. However, the basis B is only a means to define the linear isomorphism L_{B}, which is what we are really after. It allows us to pass back and forth between V and R^{n}. Sometimes, it is more convenient to specify L_{B} or L_{B}^{−1} directly and forget about the basis.

For the inverse linear transformation L_{B}^{−1}: V → R^{n}, we clearly have L_{B}^{−1}(b_{i}) = e_{i}. More generally, we convert any vector in V to an n-tuple of numbers.

Definition 5 Given a vector v ∈ V, its coordinate vector with respect to the basis B is the vector

[v]_{B} = L_{B}^{−1}(v) in R^{n}.  (6)

Example If V = R^{n} and we choose the standard basis E, L_{E} is the identity natural transformation and we have [x]_{E} = x. (But if we choose a different basis B, [x]_{B} ≠ x in general.

Change of basis Of course the coordinate vector [v]_{B} depends on the choice of basis B. The first basis chosen may not be the best. Let C = {c_{1}, c_{2}, …, c_{n}} be another basis of V. Then L_{B}^{−1} ^{∘} L_{C} is a linear transformation from R^{n} to R^{n}, and by Theorem 1(a), it has a matrix.

Definition 7 The transition matrix from C to B is the matrix P of the linear transformation L_{B}^{−1} ^{∘} L_{C}: R^{n} → R^{n}.

To find P explicitly, we use equations (2) and (6) to compute

P e_{i} = L_{B}^{−1}(L_{C}(e_{i})) = L_{B}^{−1}(c_{i}) = [c_{i}]_{B}.

Then equation (3) yields immediately

P = [[c_{1}]_{B}|[c_{2}]_{B}| … |[c_{n}]_{B}].  (8)

Thus the columns of P express the new basis vectors in terms of the old basis; note the direction.

Theorem 9 Let B and C be bases of V, and P be the transition matrix from C to B. Then

[v]_{B} = P [v]_{C}  in R^{n}.  (10)

Proof This equation translates the trivial statement

L_{B}^{−1}(v) = L_{B}^{−1}(L_{C}(L_{C}^{−1}(v))) = (L_{B}^{−1} ^{∘} L_{C})(L_{C}^{−1}(v)) in R^{n}

using equations (2) and (6).

So P is the matrix we need to transform new coordinates to old ones. We often need the reverse direction, to convert old coordinates to new coordinates.

Lemma 11 If P is the transition matrix from the basis C to the basis B, then the transition matrix from B to C is its inverse, P^{−1}.

Proof Theorem 1(b) shows that P^{−1} is the matrix of the linear transformation

(L_{B}^{−1} ^{∘} L_{C})^{−1} = L_{C}^{−1} ^{∘}(L_{B}^{−1})^{−1} = L_{C}^{−1} ^{∘} L_{B}: R^{n} → R^{n}.

By Definition 7, this is just the matrix we want.

Alternatively, we can simply multiply equation (10) on the left by P^{−1}.

Inner product spaces If V is an inner product space, we want to take advantage of the extra structure and choose not just any basis.

Theorem 12 (=Thm. 6.3.2) If V is an inner product space and B is a basis of V, then L_{B} preserves the inner product structure, ⟨L_{B}(x), L_{B}(y)⟩ = x.y (and hence the norm, ∥L_{B}(x)∥ = ∥x∥), if and only if the basis B is orthonormal.

Similarly, the relevant transition matrices take a special form.

Theorem 13 Suppose B and C are orthonormal bases of V. Then the transition matrix P from C to B is an orthogonal matrix.

Proof By Theorem 12, L_{B} and L_{C} preserve the inner-product structure, and therefore so does L_{B}^{−1} ^{∘} L_{C}. By Theorem 6.5.3 (or 6.5.1), its matrix P is orthogonal.