Linear Algebra - Vector Space (set of vector)

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

There is different way to specify a vector space:

• as the null space of a matrix
• of in terms of generators

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:

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}.

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

• Linear Algebra - (Direct Sum | Union) of vector spaces

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*

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

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

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>

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.

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

Linear Algebra - Dual of a vector space

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

See trivial linear combination

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

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.

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