# Coordinates vector

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

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## Definition

Let V be a linear space with dim V = n and let

[itex] B = \{ b_1, b_2, b_3, \cdots, b_n \} [itex]

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

[itex] v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b_n [itex]

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:

[itex] [ v ]_B = \begin{bmatrix} \alpha _1 \\ \ddots \\ \alpha _n \end{bmatrix} [itex]

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

The mapping which matches each vector v from V to its coordinate vector [v]B is an isomorphism: a linear transformation which is a one-to-one correspondence and onto. This means that every finite-dimensional linear space can be treated as a "columns 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 (matrices, 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:

[itex] B_P = \{ 1, x, x^2, x^3 \} [itex]

matching

[itex] 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 [itex]

then the corresponding coordinate vector to the polynomial

[itex] p \left( x \right) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 [itex] is [itex] \begin{bmatrix} a_0 \\ a_1 \\ a_2 \\ a_3 \end{bmatrix} [itex] .

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

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

### Example 2

The Pauli matrices which represent 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, ..., bn . Then,

[itex] v = [M]^{B} [v]_B [itex].

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

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

[itex] [v]_C = [M]_{C}^{B} [v]_B [itex]

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

[itex] [M]_{C}^{B} [M]_{B}^{C} = [M]_{C}^{C} = Id [itex]
[itex] [M]_{B}^{C} [M]_{C}^{B} = [M]_{B}^{B} = Id [itex]

Remarks:

1. The basis transformation matrix can be regarded as an automorphism over V.
2. [itex] [M]_{E}^{B} = [M]^{B} [itex] where E is the standard basis.
3. In order to easily remember the theorem
[itex] [v]_C = [M]_{C}^{B} [v]_B [itex]
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 convenience.he:קואורדינטות (אלגברה)

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