Vector-Space

• Definition:
  • The set of all column n-vectors is called
    n-dimensional real space and is denoted .^[2]
  • is the component of .

• Definition:
  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.

• If V is a vector space and x ∈V , then
  1. 0x = 0
  2. x + v = 0 implies that v = -x
  3. (-1)x = -x.
• Proof
  1. x=(1+0)x=1x+0x=x+0x, therefore 0x=0
  2. x+v=0, then –x=-x+0=-x+(x+v)=v
  3. 0=0x=(1-1)x=1x+(-1)x, x+(-1)x=0, therefore (-1)x=-x

• 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)

• 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)

• 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 :
    ()={}

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

• 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

• 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}.

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

• 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.

Let be n vectors in and let .

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

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.

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

• 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

• 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.

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

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 .

• 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 .

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 {}.

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.

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

• 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.

• 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

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

• 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

• 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

• rank(A)+dim(N(A))=n

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 A spans
• The column space of A is
• Rank(A) = m
• A𝒙=𝒃 always have at least 1 solution for any 𝒃