Let V be a vector space and let S = {v₁,v₂, ¼, v_n} be a set of vectors in V. Recall that S forms a basis for V if the following two conditions hold:

1. S is linearly independent.

2. S spans V.

We will focus on vectors in R², although all of this generalizes to Rⁿ. The standard basis in R² is {[1 0]^T,[0 1]^T}. We specify other bases with reference to this rectangular coordinate system.

Let B = {u,w} and B¢ = {u¢,w¢} be two bases for R². For a vector v Î V, given its coordinates [v]_B in basis B we would like to be able to express v in terms of its coordinates [v]_B¢ in basis B¢, and vice versa.

Suppose the basis vectors u¢ and w¢ fo B¢ have the following coordinates relative to the basis B:

[ u¢ ]B = é ê ë a b ù ú û [ w¢ ]B = é ê ë c d ù ú û .

This means that u¢ = au + bw w¢ = cu + dw

The change of coordinates matrix from B¢ to B

The transition matrix P is invertible. In fact, if P is the change of coordinates matrix from B¢ to B, then P^-1 is the change of coordinates matrix from B to B¢:

Example

Let B = {[1 0]^T,[0 1]^T} and B¢ = {[3 1]^T,[-2 1]^T}. The change of basis matrix from B¢ to B is

In the following example, we introduce a third basis to look at the relationship between two non-standard bases.

Example

Let B¢¢ = ì í î é ê ë 2 1 ù ú û , é ê ë 1 4 ù ú û ü ý þ .

To find the change of coordinates matrix from the basis B¢ of the previous example to B¢¢, we first express the basis vectors

é ê ë 3 1 ù ú û = 11 7 é ê ë 2 1 ù ú û - 1 7 é ê ë 1 4 ù ú û é ê ë -2 1 ù ú û = -9 7 é ê ë 2 1 ù ú û + 4 7 é ê ë 1 4 ù ú û

Set é ê ë 3 1 ù ú û = a é ê ë 2 1 ù ú û + b é ê ë 1 4 ù ú û é ê ë -2 1 ù ú û = c é ê ë 2 1 ù ú û + d é ê ë 1 4 ù ú û and solve the resulting systems for a,b,c, and d.

Thus, the transition matrix from B¢ to B¢¢ is

Rotation of the Coordinate Axes

Suppose we obtain a new coordinate system from the standard rectangular coordinate system by rotating the axes counterclockwise by an angle q. The new basis B¢ = {u¢, v¢} of unit vectors along the x¢- and y¢-axes, respectively, has coordinates

Example

The vector [v]_B = [3 2]^T in the original coordinate system has coordinates

In the following Exploration, set up your own basis in R² and compare the coordinates of vectors in your basis to their coordinates in the standard basis.

Let B = {u,v} and B¢ = { u¢, v¢ } be two bases for R². If [u]_B = [a b]^T and [v]_B = [c d]^T, then

P =

é ê ê ê ë

a c b d

ù ú ú ú û

is the change of coordinates matrix from B¢ to B and P^-1 is the change of coordinates matrix from B to B¢. That is, for any v Î V,