Coordinate vector - Citizendia

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. Linear algebra is the branch of Mathematics concerned with In Mathematics, specifically in Linear algebra, the coordinate space, F n, is the prototypical example of an n -dimensional 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. For other uses see Abstract In Philosophy it is commonly considered that every object is either abstract or concrete In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, a column vector or column matrix is an m × 1 matrix, i

1 Definition
2 The standard representation
3 Examples
- 3.1 Example 1
- 3.2 Example 2
4 Basis transformation matrix
- 4.1 Corollary
- 4.2 Remarks

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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the dimension of a Vector space V is the cardinality (i In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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 to be the following column vector:

$[ v ]_B = \begin{bmatrix} \alpha _1 \\ \vdots \\ \alpha _n \end{bmatrix}.$

This is also called the representation of v with respect of B, or the B representation of v. In Linear algebra, a column vector or column matrix is an m × 1 matrix, i

The α-s are called the coordinates of v. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point

The standard representation

We can mechanize the above transformation by defining a function $φ B$ , called the standard representation of V with respect to B, that takes every vector to its coordinate representation: $φ B (v) = [v] B$ . Then $φ B$ 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 $φ B$ to be its inverse. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B

Examples

Example 1

Let P3 be the space of all the algebraic polynomials in degree less than 4 (i. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations 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. In Mathematics, an operator is a function which operates on (or modifies another function In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

Example 2

The Pauli matrices which represent the spin operator when transforming the spin eigenstates into vector coordinates. The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

Basis transformation matrix

Let B and C be two different bases of a vector space V, and let's mark with $[M]_{C}^{B}$ the matrix which has columns consisting of the C representation of basis vectors b₁, b₂, . In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally . . , b_n:

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

This matrix is referred to as the basis transformation matrix from B to C, and can be used for transforming any vector v from a B representation to a C representation, according to the following theorem:

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

If E is the standard basis, the transformation from B to E can be represented with the following simplified notation:

$v = [M]^B [v]_B. \,$

where

$v = [v]_E, \,$ and

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

Corollary

This matrix is Invertible matrix and M^-1 is the basis transformation matrix from C to B. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In Mathematics, the standard basis (also called natural basis or canonical basis) of the n- dimensional Euclidean space In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- 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

The basis transformation matrix can be regarded as an automorphism over V. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself
In order to easily remember the theorem

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

notice that M 's superscript and v 's subscript indices are "canceling" each other and M 's subscript becomes v 's new subscript. This article is about the terms 'subscript' and 'superscript' as used in typography This article is about the terms 'subscript' and 'superscript' as used in typography This "canceling" of indices is not a real canceling but rather a convenient and intuitively appealing manipulation of symbols, permitted by an appropriately chosen notation.

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