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Bases and Linear mapping - Coggle Diagram
Bases and Linear mapping
DEF (subspace):
V vector space over F (field)
U c V a subset
U is a subspace of V
IF the following HOLD:
(S1) ∀ u1, u2 ∈ U => u1+u2 ∈ U
(S2) ∀ c ∈ F, ∀ u ∈ U => c u ∈ U
DEF (finitely-generated):
a vector space V is called
finitely-generated
IF ∃ {v1,...,vr) c V s.t.
L{v1,...,vr}=V
Note: Rn[x] = L{1,x,...,x^n} is finitely generated
DEF (Linearly independent):
{v1,...,vr} c V, V is a vector space, {v1,...,vr} is L.I.
IFF: a1v1+...+arvr = 0 => a1 = ... = ar = 0
DEF (basis):
V is a vector space over F,
{v1,...,vr} c V is called a Basis of V
IF B1 and B2 hold:
(B1) L{v1,...,vr} = V
(fin-gen)
(B2) {v1,...,vr} is L.I.
Note: Rn[x] has basis {1,x,...,x^n}
canonical basis of R3: {(1,0,0)=e1, (0,1,0)=e2, (0,0,1)=e3}
DEF (Linear Mapping):
V,W vector spaces over F, a mapping f: V -> W
is called a Linear Mapping
IF the following hold:
(LM1) ∀ v1,v2 ∈ V => f(v1+v2) = f(v1) + f(v2)
(LM2) ∀ c ∈ F, ∀ v ∈ V => f(cv) = c f(v)
Note: if f is L.M. then: f(0v) = 0w
DEF (Injectivity and Surjectivity):
f: V -> W is said to be
injective or into IF: f(v1) = f(v2) => v1 = v2
Surjective or onto IF: ∀ w ∈ W, ∃ v ∈ V s.t. f(v) = w
f a L.M. injective + surjective it is called an ISOMOPHISM
Lazy Lemma:
Given f: V -> W a linear map, THEN f is injective
IFF f(v) = 0w => v=0v
(we prove that since f(0v) goes to 0w then 0v is the only element going to 0v)
f injective => f(v1) = f(v2) => v1 = v2
f injective L.M. => f(v) = 0w iff v = 0v
THEOREM:
V a vector space with a basis having n elements, then:
(i) IF {v1,...,vm} c V is L.I. then m <= n
(ii) IF L{v1,...,vp} = V then p >= n
Ex 1:
4 vector in R3 are not L.I. but L{v1,v2,v3,v4} = R3 [p>=n]
EX 2:
2 vectors in R3 do not generate R3 (L{v1,v2} =! R3) but they may be L.I. [m<=n]
Corollary:
V vector sapce with basis n, then all basis have n elements
Ex 3:
dimensions:
Rn[x] has dim = n+1
Rd has dim = d
Rn,m has dim = n * m
Lemma:
A ∈ Rm,n then
f: Rn,1 -> Rm,1
x -> Ax
is a linear map
see examples:
f: R2 -> R3
(x,y) -> (x+2y, 3x+4y, 5x+6y)
f(x,y) = A(x,y) where A = ((1,2),(3,4),(5,6))
Lemma:
V,W vector spaces {v1,...,vn} a basis of V
to assign a linear map f: V -> W it isenough to assign f(v1),...,f(vn)
DEF (associated matrix):
g: V -> W a linear map
{v1,...,vn} basis of V
{w1,...,wm} basis of W
the associated matrix to g w.r.t. the 2 given bases is M(g) mxn matrix
Note: IF f(x) = Ax
f: R^n -> R^m
THEN A is the associated matrix w.r.t. canonical bases