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# Introduction

Definition 1: Vector space
A vector space $S S$ is a collection of "vectors" such that (1) if $f 1 ∈S⇒α⁢ f 1 ∈S f 1 S α f 1 S$ for all scalars $αα$ (where $α∈R α$, $α∈C α$, or some other field) and (2) if $f 1 ∈S f 1 S$, $f 2 ∈S f 2 S$, then $( f 1 + f 2 )∈S f 1 f 2 S$
To define an vector space, we need
• A set of things called "vectors" ($X X$)
• A set of things called "scalars" that form a field ($A A$)
• A vector addition operation ()
• A scalar multiplication operation ($* *$)
The operations need to have all the properties of given below. Closure is usually the most important to show.

# Vector Spaces

If the scalars $αα$ are real, $SS$ is called a real vector space.

If the scalars $αα$ are complex, $SS$ is called a complex vector space.

If the "vectors" in $SS$ are functions of a continuous variable, we sometimes call $SS$ a linear function space

# Properties

We define a set $V V$ to be a vector space if

1. $x+y=y+x x y y x$ for each $x x$ and $y y$ in $V V$
2. $x+(y+z)=(x+y)+z x y z x y z$ for each $x x$, $y y$, and $z z$ in $V V$
3. There is a unique "zero vector" such that $x+0=x x 0 x$ for each $x x$ in $V V$ (0 is the field additive identity)
4. For each $x x$ in $V V$ there is a unique vector $−x x$ such that $x+−x=0 x x 0$
5. $1⁢x=x 1 x x$ (1 is the field multiplicative identity)
6. $( c 1 ⁢ c 2 ) ⁢x= c 1 ⁢ ( c 2 ⁢ x ) ( c 1 ⁢ c 2 ) x c 1 ( c 2 ⁢ x )$ for each $x x$ in $V V$ and $c 1 c 1$ and $c 2 c 2$ in $ℂ ℂ$
7. $c⁢(x+y)=c⁢x+c⁢y c x y c x c y$ for each $x x$ and $y y$ in $V V$ and $c c$ in $ℂ ℂ$
8. $( c 1 + c 2 )⁢x= c 1 ⁢x+ c 2 ⁢x c 1 c 2 x c 1 x c 2 x$ for each $x x$ in $V V$ and $c 1 c 1$ and $c 2 c 2$ in $ℂ ℂ$

# Examples

• $Rn=real vector space n real vector space$
• $Cn=complex vector space n complex vector space$
• $L 1 ⁢R= f⁢t f⁢t ∫−∞∞|f⁢t|dt<∞ L 1 f t t f t f t$ is a vector space
• is a vector space
• is a vector space
• $ℓ 1 ⁢Z ℓ 1$, $ℓ 2 ⁢Z ℓ 2$, $ℓ ∞ ⁢Z ℓ ∞$ are vector spaces
• The collection of functions piecewise constant between the integers is a vector space

• $ℝ + 2= x 0 x 1 x 0 x 1 ( x 0 >0)∧( x 1 >0) ℝ + 2 x 0 x 1 x 0 0 x 1 0 x 0 x 1$ is not a vector space. $11∈ ℝ + 2 1 1 ℝ + 2$, but $∀α,α<0:α⁢11∉ ℝ + 2 α α 0 α 1 1 ℝ + 2$
• $D=∀ z ,|z|≤1:z∈C D z z 1 z$ is not a vector space. $( z 1 =1)∈D z 1 1 D$, $( z 2 =i)∈D z 2 D$, but $( z 1 + z 2 )∉D z 1 z 2 D$, $| z 1 + z 2 |=2>1 z 1 z 2 2 1$