Basis transformation matrix

Coordinates vector is a very important concept in Linear algebra and representation theory.

Definition

Let V be linear space with dimV = n and let

<math> B = \{ b_1 , b_2 , b_3 , \cdots , b_n \} <math>

be a linear basis for V. Therefore for every <math> v \in V <math> there is a linear combination (which is unique to v) of the basis vectors such as

<math> v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b_n <math>

The α-s are determined uniquely by v and B (the theorem of basis guarantees this) and therefore we can say that the following is a Representation of v in the B basis. Now, we define the coordinates vector of v according to B (also called B representation of v) by:

<math> [ v ]_B = \begin{bmatrix} \alpha _1 \\ \ddots \\ \alpha _n \end{bmatrix} <math>

and the α-s are called the coordinates of v.

The mapping which math each vector v from V to its coordinate vector [v]_B is an isomorphism: a linear transformation which is one-to-one correspondance and onto. This means that every finit-dimensional linear space can be treated as a "collums and squares" space Fⁿ where n is the dimension of V and F is the field on which V is defined.

Coordinates vector is a very important concept in Linear algebra and representation theory, since it allows every calculation with abstract objects to be transformed into a calculation with blocks of numbers (matrixes, column vectors) which we know how to do explicitly.

Example 1

Let P3 be the space of all the algebraic polynomials in degree less than 4 (i.e. the highest exponent of x can be 3). This space is linear and spanned by the following polynomials:

<math> B_P = \{ 1 , x , x^2 , x^3 \} <math>

matching

<math> 1 := \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \quad ; \quad x := \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \quad ; \quad x^2 := \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \quad ; \quad x^3 := \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \quad <math>

then the corresponding coordinate vector to the polynomial

<math> p \left( x \right) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 <math> is <math> \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \end{bmatrix} <math> .

According to that representation, the differentiation operator d/dx which we shall mark D will be represented by the following matrix:

<math> Dp(x) = P'(x) \quad ; \quad [D] =

\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} <math> Using that method it is easy to explore the properties of the operator: such as inveritiblity, hermitian or anti-hermitian or none, spectrum and eigenvalues and more.

Example 2

The Pauli matrices which represents the spin operator when transforming the spin eigenstates into vector coordinates.

Basis transformation matrix

Let's mark with [M]^B the matrix which has columns consisting of b₁ , b₂ , ... , b_n . Then,

<math> v = [M]^{B} [v]_B <math>.

This formaliam can be generalized for transforming v from B representation to a C representation (where C is another basis). Defining basis tramsformation matrix from B to C as the following matrix:

<math> [M]_{C}^{B} = \begin{bmatrix} \ [b_1]_C & \cdots & [b_n]_C \ \end{bmatrix} <math>

we receive the following theorem:

<math> [v]_C = [M]_{C}^{B} [v]_B <math>

Corolary:
This matrix is Invertible matrix and M^-1 is the basis transformation matrix from C to B. In other words,

<math> [M]_{C}^{B} [M]_{B}^{C} = [M]_{C}^{C} = Id <math>

<math> [M]_{B}^{C} [M]_{C}^{B} = [M]_{B}^{B} = Id <math>

Remarks:

1. The basis transformation matrix can be regarded as an automorphism over V.
2. <math> [M]_{E}^{B} = [M]^{B} <math> where E is the standard basis.
3. In order to easily remember the theorem

<math> [v]_C = [M]_{C}^{B} [v]_B <math>

notice that the M's sup-index and v's sub-index are "canceling" each other and the M's sub-index is what remains and become v's new sub-index. The "canceling" of index is not a real canceling but rather a manipulation of symbols which serves us for purposes of conveniency.