MATH547DS L03: Formal Rules

Overview

• Algebraic structures

• Vector subspaces

Algebraic structures

• Groups

  Addition rules $a+b\in G$ Closure $a+(b+c)=(a+b)+c$ Associativity $0+a=a$ Identity element $a+(-a)=0$ Inverse element

  Table 1. Group $𝒢=(G,+)$ properties, for $\forall a,b,c\in G$

• Rings

  Addition rules $(R,+)$ is a commutative (Abelian) group Multiplication rules $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ Associativity $a\cdot 1=1\cdot a=a$ Identity element Distributivity $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ on the left $(a+b)\cdot c=(a\cdot c)+(b\cdot c)$ on the right

  Table 2. Ring $ℛ=(R,+,\cdot )$ properties, for $\forall a,b,c\in R$.

Algebraic structures

• Fields

  Addition rules $(R,+)$ is a commutative (Abelian) group Multiplication rules $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ Associativity $a\cdot 1=1\cdot a=a$ Identity element Distributivity $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ on the left $(a+b)\cdot c=(a\cdot c)+(b\cdot c)$ on the right

  Table 3. Ring $ℛ=(R,+,\cdot )$ properties, for $\forall a,b,c\in R$.

Vector subspaces

Definition. (Vector Subspace) . $𝒰=(U,S,+,\cdot )$ with $U\ne \varnothing$ is a vector subspace of vector space $𝒱=(V,S,+,\cdot )$ over the same field of scalars S if $U\subseteq V$ and $\forall a,b\in S$, $\forall 𝒖,𝒗\in U$, the linear combination $a𝒖+b𝒗\in U$.

1. Column space, $C\left(𝑨\right)=\{𝒃\in {ℝ}^m|\exists 𝒙\in {ℝ}^n\mathrm{such}\mathrm{that}𝒃=𝑨𝒙\}\subseteq {ℝ}^m$, the part of ${ℝ}^m$ reachable by linear combination of columns of $𝑨$

2. Left null space, $N\left({𝑨}^T\right)=\{𝒚\in {ℝ}^m|{𝑨}^T𝒚=0\}\subseteq {ℝ}^m$, the part of ${ℝ}^m$ not reachable by linear combination of columns of $𝑨$

3. Row space, $R\left(𝑨\right)=C\left({𝑨}^T\right)=\{𝒄\in {ℝ}^n|\exists 𝒚\in {ℝ}^m\mathrm{such}\mathrm{that}𝒄={𝑨}^T𝒚\}\subseteq {ℝ}^n$, the part of ${ℝ}^n$ reachable by linear combination of rows of $𝑨$

4. Null space, $N\left(𝑨\right)=\{𝒙\in {ℝ}^n|𝑨𝒙=0\}\subseteq {ℝ}^n$, the part of ${ℝ}^n$ not reachable by linear combination of rows of $𝑨$