Coordinate vector

In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fⁿ. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers (matrices and column vectors), which we know how to do explicitly.

Definition

Let V be a vector space of dimension n over a field F and let

$B=\{b_{1},b_{2},\ldots ,b_{n}\}$

be an ordered basis for V. Then for every $v\in V$ there is a unique linear combination of the basis vectors that equals v:

$v=\alpha _{1}b_{1}+\alpha _{2}b_{2}+\cdots +\alpha _{n}b_{n}$

By one of the defining properties of bases, the α-s are determined uniquely by v and B. Now, we define the coordinate vector of v relative to B (also called the B representation of v) to be the following column vector:

$[v]_{B}={\begin{bmatrix}\alpha _{1}\\\vdots \\\alpha _{n}\end{bmatrix}}.$

The α-s are called the coordinates of v.

The standard representation

We can mechanize the above transformation by defining a function $\phi _{B}$, called the standard representation of V with respect to B, that takes every vector to its coordinate representation: $\phi _{B}(v)=[v]_{B}$. Then $\phi _{B}$ is a linear transformation from V to Fⁿ. In fact, it is an isomorphism, and its inverse $\phi _{B}^{-1}:\mathbf {F} ^{n}\to V$ is simply

$\phi _{B}^{-1}(\alpha _{1},\ldots ,\alpha _{n})=\alpha _{1}b_{1}+\cdots +\alpha _{n}b_{n}.$

Alternatively, we could have defined $\phi _{B}^{-1}$ to be the above function from the beginning, realized that $\phi _{B}^{-1}$ is an isomorphism, and defined $\phi _{B}$ to be its inverse.

Examples

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:

$B_{P}=\{1,x,x^{2},x^{3}\}$

matching

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

then the corresponding coordinate vector to the polynomial

$p\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$ is ${\begin{bmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\end{bmatrix}}$ .

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

$Dp(x)=P'(x)\quad ;\quad [D]={\begin{bmatrix}0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\\\end{bmatrix}}$

Using that method it is easy to explore the properties of the operator: such as invertibility, 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, ..., b_n . Then,

$v=[M]^{B}[v]_{B}$.

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:

$[M]_{C}^{B}={\begin{bmatrix}\ [b_{1}]_{C}&\cdots &[b_{n}]_{C}\ \end{bmatrix}}$

we receive the following theorem:

$[v]_{C}=[M]_{C}^{B}[v]_{B}$

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

$[M]_{C}^{B}[M]_{B}^{C}=[M]_{C}^{C}=\mathrm {Id}$

$[M]_{B}^{C}[M]_{C}^{B}=[M]_{B}^{B}=\mathrm {Id}$

Remarks:

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

$[v]_{C}=[M]_{C}^{B}[v]_{B}$

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.