Vector Space

2024-09-24 1
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Vector and Vector Space Axioms

[1]

Definition

The set of all column n-vectors is called N-dimensional real space and is denoted .[2]

  • e.g. is the component of .

Addition and Scalar Multiplication

  • Let and be two elements of and let be a scalar. Addition and scalar multiplication are defined as
    • Addition:
    • Scalar Multiplication:

Prove a VectorSpace

  • ( is closed under addition) 加法闭合
  • ( is closed under scalar multiplication) 乘法闭合

Zero Vector

Zero Vector Definition:

The vector , having n zero components, is called the zero vector of and is denoted .
Property: .

Negative Vector

Negative Vector Definition:

The vector is written and is called the negative of .
It is a vector having the same magnitude as , but lies in the opposite direction to .
Property: for any vector ,

Vector Space Axioms

Vector Space Axioms

Let V be a set on which the operations of addition and scalar multiplication are defined.[3]
The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied:

  • . for any x and y in V.
  • . for any x, y, and z in V.
  • . There exists an element 0 in V such that x + 0 = x for each x ∈ V.
  • . For each x ∈ V, there exists an element −x in V such that x + (−x) = 0.
  • for each scalar α and any x and y in V.
  • for any scalars α and β and any x ∈ V.
  • for any scalars α and β and any x ∈ V.
  • for all x ∈ V.

is a Vector Space [4]

  • :
    • Addition: The sum is still an matrix
    • Scalar multiplication: It is still an matrix

is a Vector Space

Definition:

is the set of all real-valued continuous functions defined on the interval .
Vectors in this case are continuous real-valued functions.

  • :
    • Addition:
    • Scalar multiplication:

is a Vector Space

Definition:

A polynomial of degree k (k≥0) is a function of the form

  • where .
  • is also considered to be a polynomial.

Vector Space Theorem

If is a vector space and then

Subspace & Null space

Subspaces of Vector Spaces

Subspaces Definition:

Let V be a vector space (for example, ). A nonempty subset S of V is a subspace if it is closed under addition and closed under scalar multiplication.

To show that a subset S of a vector space forms a subspace, we must show that

  • S is nonempty (it contains the zero vector)
  • (S is closed under addition)
  • (S is closed under scalar multiplication)

Trivial subspace

  • Definition:
    If V is a vector space, then V is a subspace of itself, and S = {0} is a subspace of V.
    All the other subspaces are referred to as proper subspaces.
  • We refer to {0} as the zero subspace.

1.2.2. Null space of a Matrix

  • Definition:
    For a m×n matrix A, let 𝑁(A) denote the set of all solutions to the homogeneous system Ax = 0, that is 𝑁(A)={|Ax=𝟎}
    Null space is a subspace
    • Addition: if 𝒙,𝒚∈𝑁(A), then A𝒙=𝟎, A𝒚=𝟎⇒A(𝒙+𝒚)=𝟎, so x+𝑦∈𝑁(A)
    • Scalar multiplication: if 𝒙∈𝑁(A), c∈R, then A𝒙=𝟎⇒A(c𝒙)=𝟎 ⇒ c𝒙∈𝑁(A)

1.2.3. The Span of a Set of Vectors

  • Definition:
    • For vectors from a vector space V and scalars the sum is called a linear combination of .
    • The set of all possible linear combinations of is called the span of , denoted by :
      ()={}

1.2.4. Theorem

If are vectors from a vector space V , then is a subspace of V.

1.2.5. Spanning Set

  • Definition:
    The set of vectors {} is a spanning set for a vector space V if and only if every vector in V can be written as a linear combination of , that is to say, for any x𝜖V there exist scalars such that

How to check if a set spans

  • Step 1: Put the column vectors into a matrix, A=[,...]
  • Step 2: Augment the matrix by the vector
  • Step 3: Row reduce the augmented matrix to check whether it is consistent for All a, b, c
  • Step 4: If is consistent for All a, b, c, then the columns of , is a spanning set of .

1.3. Linear Independence

1.3.1. Linear Dependence

  • Definition:
    The vectors in a vector space V are said to be linearly dependent (线性相关) if and only if there exist scalars , not all zero, such that
  • Let , then is linearly dependent if and only if the null space of A, 𝑁(A), is NOT the zero subspace {0}.

1.3.2. Theorem 1.3.0

Nonzero vectors are linearly dependent if and only if at least one vector is a linear combination of the others.

1.3.3. Linear Independence

  • Definition:
    The vectors in a vector space V are said to be linearly independent (线性无关) if implies that all the scalers must equal to 0.
  • Let , then is linearly independent if and only if the null space of A is the zero subspace: 𝑁(A)={𝟎}.
  • None of the vectors can be written as a linear combination of the others.

1.3.4. Theorem 1.3.1 [5]

Let be n vectors in and let .

  • The vectors will be linearly dependent if and only if 𝑋 is singular.

1.3.5. Theorem 1.3.2

Let be vectors in a vector space V.

  • A vector can be written uniquely as a linear combination of if and only if are linearly independent.

1.4. Basis and Dimension

  • Definition：
    The vectors form a basis for a vector space V if and only if
    • are linearly independent.
    • span V.
  • If form a basis for V, then:for any 𝒃∈V, there exists unique such that

1.4.1. Standard Basis of

  • Definition：
    The set {} of n vectors is the standard basis for .
    • Standard basis for : {}.
    • Standard basis for : {}.

1.4.2. Dimension

  • Definition:
    Let V be a vector space. If V has a basis consisting of n vectors, we say that V has dimension n and write dim V=n
  • The subspace {0} of V is said to have dimension 0.
  • V is finite dimensional if there is a finite set of vectors that spans V; otherwise V is infinite dimensional.
    • dim R^n = n
    • dim R^(m×n) = m×n
    • dim Pn = n

1.4.3. Theorem 1.4.1

  • If S = {} is a spanning set for a vector space V , then any collection of m vectors in V with m > n is linearly dependent.

Important Corollary (1.4.2)

  • If {} and {} are both bases for a vector space V , then m = n.

1.4.4. Theorem 1.4.3

If V is a vector space with dim ⁡V = n > 0, then

  • Any set of n linearly independent vectors also spans V;
  • Any n vectors that span V are also linearly independent

1.4.5. Theorem 1.4.4

If V is a vector space with dim V = n > 0, then

  • No set with k < n vectors can span V ;
  • Any subset of k < n linearly independent vectors can be extended to form a basis for V;
  • Any spanning set of V with m > n vectors can be cut down to form a basis for V .

1.5. Change of Basis

1.5.1. Coordinate Vector

  • Definition：
    Let V be a vector space and let E={} be an ordered basis for V. If v is any element of V, then v can be written uniquely in the form
    where are scalers.
    Thus, we can associate each vector v a unique column vector {}. The vector is called the coordinate vector of v with respect to (w.r.t). the ordered basis E, denoted .

1.5.2. Changing Coordinates

Given vector , find its coordinates (w.r.t). and .

  • As we have x=Uc.
  • Matrix U is nonsingular, therefore .
  • Thus, is the transition matrix from {} to {}.

Given vector , find its coordinates w.r.t. and .

  • If c is the coordinate vector of x with respect to the basis {}, then
    • x=Uc
  • And
  • Where U={} is called the transition matrix from the {} to the standard basis {}, and is the transition matrix from {} to {}.

Changing coordinate vectors from one basis {} of to another basis {}:

  • Suppose for a given vector x, its coordinates (w.r.t.) {} are known: We want to write x as .
  • We have ; Let and .
  • Then is the transition matrix from {} to {}.
  • Property
    • A transition matrix is nonsingular
    • If S is the transition matrix from {} to {}, then is the transition matrix from {} to {}.

1.6. Row space & column space

1.6.1. Row space & column space

For an m×n matrix A, the rows are n-vectors from , and the column vectors are m-vectors from .

  • Definition:
    • For an m×n matrix A, the subspace of spanned by the row vectors of A is called the row space of A.
    • The subspace of spanned by the column vectors of A is called the column space of A.

1.6.2. Theorem 1.6.1

  • Two row-equivalent matrices have the same row space.

1.6.3. Theorem 1.6.2

  • Two row-equivalent matrices have the same null space.
  • If U is the rref of A and , then als .
  • All the column vectors of A corresponding to the leading variables in U form a basis of the column space of A.

1.6.4. Rank

  • Definition:
    The rank of a matrix A is the dimension of the row space of A, denoted rank(A).
  • Property: The rank of a reduced row echelon form is equal to the number of lead variables

1.6.5. Dimension Theorem 1.6.3

  • dim(Row Space A) = dim(Column Space A) = rank(A).

1.6.6. Consistency theorem for Linear System 1.6.4

  • The linear system Ax = 𝐛 is consistent if and only if 𝐛 is in the column space of A.
    • An n×n square matrix A is nonsingular if and only if the column vectors of A form a basis for ⇔ Ax=b always have unique solution ⇔ 𝑟ank(A)=n⇔det⁡(A)≠0

1.6.7. Nullity

  • Definition:
    The nullity of a matrix A is the dimension of the null space of A, that is nullity(A) = dim(𝑁(A)).
  • Properties：
    • The number of lead variables: rank(A)
    • The number of free variables: dim(N(A))
    • The number of variables: n

1.6.8. Theorem 1.6.5

  • rank(A)+dim(N(A))=n

1.6.9. Property

Let . The following are equivalent:

  • The columns of A are linearly independent
  • 𝑁(A)={𝟎}
  • Nullity(A) = 0
  • A𝒙=𝒃 has at most 1 solution for any 𝒃

Let . The following are equivalent:

  • The columns of spans
  • The column space of is
  • Rank(A) = m
  • always have at least 1 solution for any 𝒃
  1. Euclidean Vector Spaces↩︎
  2. Perhaps the most elementary vector spaces are the Euclidean vector spaces .↩︎
  3. By this we mean that, with each pair of elements x and y in V, we can associate a unique element x+y that is also in V, and with each element x in V and each scalar α, we can associate a unique element αx in V.↩︎
  4. Here vector is one matrix↩︎
  5. Only Square matrix↩︎
Table Of Contents
Vector Space
Vector and Vector Space Axioms
[1]
Addition and Scalar Multiplication
Prove a VectorSpace
Zero Vector
Negative Vector
Vector Space Axioms
is a Vector Space [4]
is a Vector Space
is a Vector Space
Vector Space Theorem
Subspace & Null space
Subspaces of Vector Spaces
Trivial subspace
1.2.2. Null space of a Matrix
1.2.3. The Span of a Set of Vectors
1.2.4. Theorem
1.2.5. Spanning Set
How to check if a set spans
1.3. Linear Independence
1.3.1. Linear Dependence
1.3.2. Theorem 1.3.0
1.3.3. Linear Independence
1.3.4. Theorem 1.3.1 [5]
1.3.5. Theorem 1.3.2
1.4. Basis and Dimension
1.4.1. Standard Basis of
1.4.2. Dimension
1.4.3. Theorem 1.4.1
Important Corollary (1.4.2)
1.4.4. Theorem 1.4.3
1.4.5. Theorem 1.4.4
1.5. Change of Basis
1.5.1. Coordinate Vector
1.5.2. Changing Coordinates
1.6. Row space & column space
1.6.1. Row space & column space
1.6.2. Theorem 1.6.1
1.6.3. Theorem 1.6.2
1.6.4. Rank
1.6.5. Dimension Theorem 1.6.3
1.6.6. Consistency theorem for Linear System 1.6.4
1.6.7. Nullity
1.6.8. Theorem 1.6.5
1.6.9. Property