specifically midterm Flashcards
VECTOR SPACE
∃0∈V : 0+u=u, ∀u∈V
u+v=v+u, ∀u,v∈V
u(v+w) = (u+v)+w, , ∀u,v,w∈V
∃z∈V : u+z=0, ∀u∈V
1.u=u, ∀u∈V
a.b.u = ab.u, ∀u∈V, ∀a,b∈F
a.(u+v) = (a.u)+(a.v), ∀u,v∈V, ∀a∈F
u.(a+b) = (a.u)+(b.u), ∀u∈V, ∀a,b∈F
span(U ∪ W),
U, W⊆V
= (U + W) ⊆ V
DIRECT SUM
U,W ⊆ V: U ∩ W = ∅
⟹
U⊕W bijective (every element unique)
any list of vectors containing the zero vector and/or a repeated vector is linearly dependent
DIMENSION MATRIX
A = [a1 … an] ∈ F^(mxn)
β=a1,…,an
⟹
dim span β = dim col A =: rank A
U, W subspaces of V, dimV<∞
⟹
dim U + dim W =
dim(U ∩ W) + dim (U + W)
- U, W subspaces of V, dimV<∞
- dimU + dimV >= dim V + k, k>0
⟹
U ∩ W contains k linearly independent vectors
A, B, C ∈ Mn(F): AB=I=BC ⟹
A=C
LINEAR TRANSFORMATION INDUCED BY A
T_A: F^(n) → F^(n):
x↦Ax, A∈Mmxn(F)
Ker(T)
T:V->W
= {v∈V|T(v) = 0}
Null
Ran(T)
T:V->W
= {w∈W|∃v ∈V : T(v)=w}
(image)
Col
LINEAR TRANSFORMATION PROPERTIES
- T(cv+u) = cT(v) + T(u)
- T(0) = 0
β = v1,…, vn basis for V, T∈L(V,W)
v = c1v1+…cnvn
⟹
- Tv = c1Tv1+…cnTvn
- RanT = span(Tv1,…,Tvn}
- dimRanT <= n
A∈Mn(F) invertible ⟺
- rankA=n
- columns of A form basis for Fn
β = v1,…, vn basis for V, dimV=n>0
S∈Mn(F) invertible
⟺
∃γ basis for V: S= γ[I]β
- T∈L(V)
⟹
(γ[T]γ) = S(β[T]β)S^(-1)
A, B∈Mn(F) “similar over F”
∃S∈Mn(F): A=SBS^(-1)
(equivalence relation)
A,B∈Mn(F) similar over F ⟺
- A=β[T]β and B= γ[T]γ,
where β, γ bases for some V: dimV=n - ∃λ∈F: (A-λI) similar to (B-λI)
- A-λI similar to B-λI, ∀λ∈F
A,B∈Mn(F) similar
⟹
- TrA=TrB
- detA=detB
DIMENSION THEOREM:
FOR LINEAR TRANSFORMATIONS
FOR MATRICES
dim ker T + dim ran T = dim V
dim null A + dim col A = n
dimV=dimW, T∈L(V,W) ⟹
n=m, A∈Mmxn(F) ⟹
kerT=∅ ⟺ ranT=W
nullA= ∅ ⟺ colA=Fn
INNER PRODUCT on V (5)
- <v,v> real >=0
- <v,v> = 0 ⟺ v=0
- <u+v,w> = <u,w> + <v,w>
- <cu,v> = c<u,v>
- <u,v> = <v,u>__
ORTHOGONAL PROPERTIES (3)
- u⊥v ⟺ v⊥u
- 0⊥u, ∀u∈V
- v⊥u, ∀u∈V ⟹ v=0
<u,v> = <u,w>, ∀u∈V ⟹
v=w
NORM DERIVED FROM INNER PRODUCT
||v|| = √<v,v>
“norm on V”
NORM properties
- ||u|| real >=0
- ||u||=0 ⟺ u=0
- ||cu|| = |c|||u||
- ||u+v||<= ||u||+||v||
* equality iff u,v linearly dependent and same sign
L1 norm
||u||1 = |u1| + … + |un|
(not derived from inner product)
L∞ norm
||u||∞ = max{|ui|: 1<=i<=n}
(not derived from inner product)
DERIVED NORM PROPERTIES
same as norm plus:
“pythagorean theorem”
<u,v>=0
⟹
||u+v||^2 = ||u||^2 + ||v||^2
“parallelogram identity”
||u+v||^2 + ||u-v||^2
=
2||u||^2 + 2||v||^2
“cauchy schwarz inequality”
|<u,v>| <= ||u||||v||
“polarisation identities”
<u,v> =
1/4 (||u+v||^2 - ||u-v||^2 + i||u+iv||^2 - i||u-iv||^2)
* i terms (2nd line) ignored if F=R
EUCLIDEAN INNER PRODUCT (F^n)
<x,y> = Σxi(yi)_
EUCLIDEAN NORM
||x||2 = √(Σ|xi|^2)
FROBENIUS INNER PRODUCT (Mn(F))
<A,B> = tr(A^(H)B) = Σaij(bij)_
FROBENIUS NORM
||A||2 = √tr(A^(H)A) = √Σ|aij|^(2)
L^2([a,b]) INNER PRODUCT
<f,g> = ∫(a,b)f(x)g(x)_dx
*when[a,b] = [-π,π], divide integral by π
L^2([a,b]) NORM
||f|| = √( ∫(a,b)|f(x)|^2dx)
