Problem of the week - Coordinates and change of basis matrix

A basis for a vector space V of dimension n is a set of n vectors with the property that every vector in the space can be expressed uniquely as a linear combination of the basis vectors. One of the useful features of a basis is that it enables to introduce the concept of coordinates, which are the coefficients of the linear combination when expressing a vector in terms of the vectors of the basis, and they are always specified relative to an ordered basis.

An important concept related to basis and coordinates is the change of basis matrix. When there are two ordered bases for the same vector space, the change of basis matrix from the first basis to the second one, is the matrix that allows us to get the coordinate vector relative to the second basis by using only the coordinate vector relative to the first one, without even knowing the bases themselves.

Understanding change of basis matrices will help you to understand some problems related to diagonalization and singular value decomposition, among other important concepts which are widely used in many fields of mathematics, physics and engineering.

To solve the problem of this week you will need to use the concepts of coordinate vector and change of basis matrix.

Given the change of basis matrix C, from basis A to basis B, both bases from R².

┌       ┐
│ 6  -5 │
│ 7  -6 │
└       ┘

and knowing that the coordinate vectors relative to the basis A for the vectors

v₁ = (1, 5) and v₂ = (2, 4) are [2, 1] and [-2, -4], respectively.

Find the vectors of the bases A and B?

The coordinate vector of a vector relative to a basis, is composed by the coefficients when expressing the vector as a linear combination of the vectors of the basis.

Let (a, b) and (c, d) the ordered vectors of base A, then, according to the coordinate vectors of v₁ and v₂:

(1, 5) = 2(a, b) + 1(c, d)

and

(2, 4) = -2(a, b) + -4(c, d)

which lead to the following equations:

2a + c = 1
2b + d = 5
-2a - 4c = 2
-2b - 4d = 4

The solution to this system of linear equations is:

a = 1
b = 4
c = -1
d = -3

Thus, the basis A is composed by the vectors (1 4) and (-1 -3).

As the change of basis matrix from the basis A to the basis B can be computed using the basis matrix of A and B (the basis matrix is the matrix where its columns are the components of the vectors of the basis. It is a shortcut to the change of basis matrix from a basis to the standard basis) as:

C = B-1A

So, to get the vectors of the basis B we need to isolate B, multiplying both terms to the left by B and to the right by C^-1:

B = AC-1

The matrix C, in this problem, is an involutory matrix (it is its own inverse), so:

B = AC

resulting in:

┌        ┐
│ -1   1 │
│  3  -2 │
└        ┘

Then, the vectors of B are (-1 3) and (1 -2).