What is the difinition of a linear space?
let V be non-empty set
with operations
+ : VxV -> V
* : RxV -> V
such that a+b ∈ V
λa ∈ V
for all a,b ∈ V, λ ∈ R
Fulfills 8 additional rules
Waht are the 8 rules in linear spaces?
Commutativity
a + b = b + a
for all a,b ∈ V
associativity
(a+b)+c = a+(b+c)
for all a,b,c ∈ V
there exists zero element
0 ∈ V
such that a+0 = a
for all a ∈ V
for every a ∈ V there exists inverse element
-a ∈ V
such that a + (-a) = 0
1a = a for all a ∈ V
λ(µa) = (λµ)a
for all λ, µ ∈ R, a ∈ V
Distributivity I
λ(a + b) = λa + λb
for all λ ∈ R, a, b ∈ V
Distributivity II
(λ + µ)a = λa + µa
Last changed2 years ago