Linear Algebra - Vector Space (set of vector)

Definition

A vector space is a subset of the set of function <math>F^D</math> representing a geometric object passing through the origin.

A vector space over a field F is any set V of vector :

  • with the addition and scalar-multiplication operation
  • satisfying certain axioms (e.g. commutate and distributive laws)
  • and the following properties P1, P2, P3.

This geometric subset of <math>F^D</math> satisfies three properties:

  • P1: They contains the zero vector 0 (The origin). A vector space always contains the zero vector.
  • P2: If subset contains v then it contains <math>\alpha.v</math> for every scalar <math>\alpha</math>
  • P3: If subset contains u and v then it contains u + v

The image of a linear function <math>f : V \rightarrow W</math> is a vector space.

Representation

There is different way to specify a vector space:

There is two natural way (Dual Representation) to specify a vector space V (of every subspace of <math>\mathbb{R}^D</math> ). It's to specify a basis in terms:

generator

of generators for V. <MATH>V = Span \{v_1, \dots , v_n\}</MATH>

Matrix equivalent to: <MATH>V = Row \begin{bmatrix} \begin{array}{r} v_1 \\ \hline \\ \vdots \\ \hline \\ v_n \end{array} \end{bmatrix} </MATH>

Computational Problem: Finding a basis of the vector space spanned by given vectors:

  • input: a list [v1, . . . , vn] of vectors
  • output: a list of vectors that form a basis for Span {v1, . . . , vn}.

homogeneous linear system

or of a homogeneous linear system whose solution set is the vector space V. <MATH>V = \text{Solution set of homogeneous linear system} \{x : a_1.x = 0, \dots, a_m.x = 0\}</MATH>

matrix equivalent to:

<MATH> V = \href{matrix#null space}{Null} \begin{bmatrix} \begin{array}{r} a_1 \\ \hline \\ \vdots \\ \hline \\ a_n \end{array} \end{bmatrix} </MATH>

Computational Problem: Finding a basis of the solution set of a homogeneous linear system

  • input: a list [a1, . . . , an] of vectors
  • output: a list of vectors that form a basis for the set of solutions to the system a1 · x = 0, . . . , an · x = 0

Operations

Direct Sum

Representation Transformation

From generators to homogeneous system

Definition:

  • Given system a1 · x = 0, . . . , am · x = 0, find generators v1, . . . , vn for solution set
  • Equivalently, given matrix A, find B such that Row B = Null A

Solution set is:

  • x, the set of vectors u such that a1 · u = 0, . . . , am · u = 0

<MATH> \underbrace{ \begin{bmatrix} \begin{array}{r} a_1 \\ \hline \\ \vdots \\ \hline \\ a_n \end{array} \end{bmatrix}}_{A} \begin{bmatrix} \begin{array}{r} \\ \\ x \\ \\ \end{array} \end{bmatrix} = \begin{bmatrix} \begin{array}{r} 0 \\ \hline \\ \vdots \\ \hline \\ 0 \end{array} \end{bmatrix} </MATH>

  • Equivalent: Given rows of a matrix A, find generators for Null A

If u is such a vector then <MATH>u · (\alpha_1.a_1 + \dots + \alpha_m.a_m) = 0</MATH> for any coefficient <math>\alpha_1, \dots, \alpha_m</math>

Two equiavalent computations where Algorithm X solves this operation:

  • rows of a matrix A → Algorithm X → generators for Null A
  • generators for a vector space V → Algorithm X → generators for dual space V*

From homogeneous system to generators

Definition

  • Given generators v1, . . . , vn, find system a1 · x = 0, . . . , am · x = 0 whose solution set equals Span {v1, . . . , vn}
  • Equivalently, given matrix B, find matrix A such that Null A = Row B

Computation:

  • generators for dual space V* → Algorithm Y → generators for original space V

A (V*)* = V. The dual of the dual is the original space. Algorithm X = Algorithm Y

Lexique

Span

The set of all linear combinations of some vectors v1,…,vn is called the span of these vectors and contains always the origin.

Generator

The generators for the set of vectors <math>V</math> are the vectors <math>v_1, \dots,v_n</math> in the following formula:

<MATH>V = Span \{v_1,\dots,v_n\}</MATH>

where <math>\{v_1,\dots,v_n\}</math> is a generating set for <math>V</math>

Dimension

The dimension of a vector space is the size of a basis for that vector space. The dimension of a vector space V is written dim V.

Basis

Lemma: Every finite set T of vectors contains a subset S that is a basis for Span T.

Dual

Type

Affine

If c is a vector and <math>V</math> is a vector space then

<math>c + V</math>

is called an affine space

Example: A plane or a line not necessarily that contain the origin

Trivial

Subspace

Let <math>\upsilon</math> and <math>\gamma</math> be a vector space, if <math>\upsilon</math> is a subset of <math>\gamma</math> then <math>\upsilon</math> is called a subspace of <math>\gamma</math> .

Dimension Lemma: If U is a subspace of W then:

  • Property D1: dim U <= dimW, and
  • Property D2: if dim U = dimW then U = W

Complementary subspace

When <math>U \href{direct_sum}{\oplus} V = W</math> , U and V are complementary subspace of W.

Example: Suppose U is a plane in <math>\mathbb{R}^3</math> . Then any line through the origin that does not lie in U is complementary subspace with respect to <math>\mathbb{R}^3</math>

For any finite-dimensional vector space W and any subspace U, there is a subspace V such that U and V are complementary.

Orthogonal complement

Let U be a subspace of W. For each vector b in W, we can write b as the following projections]]: <MATH>b = b^{||U} + b^{\perp U}</MATH> where:

  • <math>b^{||U}</math> is in U, and
  • <math>b^{\perp U}</math> is orthogonal to every vector in U.

Let V be the set <math>\{b^{\perp U} : b \in W\}</math> . V is the orthogonal complement of U in W


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