Vector Spaces

• A set of vectors over a field, F
• Consists of a set of two operations (addition and scalar multiplication so that
  • for each $x,y\in V$, there is a unique element $x+y\in V$
  • for each $a\in F,x\in V$, there is a unique element $ax\in V$
• Such That
  1. $\forall x,y\in V,x+y=y+x$ (commutativity of addition)
  2. $\forall x,y\in V,(x+y)+z=x+(y+z)$ (associativity of addition)
  3. $\exists 0\in V\text{ s.t. }x+0=x\forall x\in V$ (0 exists)
  4. $\forall x\in V,\exists y\in V,\text{ s.t. }x+y=0$ (additive inverse)
  5. $\forall x\in V,1x=x$ (1 exists)
  6. $\forall a,b\in F\text{ and }\forall x\in V,(ab)x=a(bx)$ (associativity of multiplication)
  7. $\forall a\in F\text{ and }\forall x,y\in V,a(x+y)=ax+ay$ (scalar distributivity)
  8. $\forall a,b\in F\text{ and }\forall x\in V,(a+b)x=ax=bx$ (vector distributivity)
• Elements are vectors

Example Vector Spaces

• Tuples
• Matrices
• Functions

Theorems

• $x,y,z\in V$ such that $x+z=y+z$ then $x=y$
• In any vector space, V, the following hold $\forall x\in V,\forall a\in F$
  • $0x=0$
  • $(-a)x=-(ax)=a(-x)$
  • $a0=0$