Field

  • closed umder 4 binary operations：$+, -, \times, \div$
  • R ，C，F（任意数域）

Vector Space

  • Definition
    • closed under 2 operations: addition, scalar multiplication
    • satisfied 8 axioms

Polynomial Space

$P_n={\sum^n_{j=0}a_j x^j|a_0,a_i,\dots,a_n\in R}$

Subspace

  • prove

    • zero vector
    • closed under addition and scalar multiplication

  • the intersection $\cap$ of two subspaces U and V is always a subspace

  • a union $\cup$ of two subspaces of V is not necessarily a subspace of V
    • the union of two subspaces of V is a subspace of V if and only if one of the subspace is contained in the other

Span

If a set of vectors $S=\{v_1,v_2,\dots,v_k\}\subset V$ is a subset of a vector of a vector space V over a field F, Span S is the intersection of all subspaces of V that contain S

  • if $S\ne\varnothing$, then

    $span\ S=\{\sum^k_{j=1}a_jv_j|a_1,\dots,a_k\in F,k=1,2,\dots\}$

  • span S is a subspace of V

The sum of two subspaces: X+Y

  • $S=X+Y=span\{X\cup Y\}=\{x+y|x\in X,y\in Y\}$

  • X,Y is subspaces of V, then X+Y is a subspace of V

  • Theorem:

    if $span\{S_X\}=X, span\{S_Y\}=Y$, then $span\{S_X\cup S_Y\}=X+Y$

Linear Combinations

A linear combination of vectors in a vector space V over a field F is any expression of the form

$a_1v_1+a_2v_2+\cdots+a_kv_k$

in which k is a positive integer, $v_1,\dots,v_k\in S，a_1,\dots,a_k\in F$

  • span S consists of all linear combinations of finitely many vectors in S