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
U is subspace of V (def.)
if U is subset of V that is a vector space with same vector addition and scalar multiplication as in V
U⊆V subpace of V
⟺
cu+v ∈U whenever u,v ∈U and c ∈F
null(A)
{x ∈F^n: Ax=0} ⊆F^n
columns in kernel
col(A)
{Ax: x∈F^n} ⊆F^n
set of all linear combinations (span) of the columns of A
list vs set
β = a1, a2,…, ar vs β = {a1, a2,…, ar}
span(U)
set of all linear combinations of elements of U (may be set U⊆F^n or list β=U)
span(∅)
span(u∈U)
= {0}
={cu: c∈F}
PROPERTIES OF SPAN (thm.1.4.9,10)
- span(U⊆V)⊆V
- U ⊆ span U
- U = span U ⟺ U⊆V
- span(spanU) = span U
- U ⊆W ⟹ span U ⊆ span W
U⊆V is spanning set of V
β:=list of vectors in V is spanning list of V
span(U) = V
span(β) = V
span(U ∪ W),
U, W⊆V
= (U + W) ⊆ V
DIRECT SUM
U,W ⊆ V: U ∩ W = ∅
⟹
U⊕W bijective (every element unique)
v1,…,vr LINEARLY DEPENDENT
∃c1,…,cr ∈F not all zero:
c1v1+…+crvr=0
v∈V linearly dependent ⟺
v=0
any list of vectors containing the zero vector and/or a repeated vector is linearly dependent
β = v1,…,vr linearly dependent ⟹
v1,…,vr, v linearly dependent, ∀v∈V
β = v1,…,vr linearly independent
c1v1+…+crvr=0 ⟺ c1=…=cr=0
v1,…,vr linearly independent
then a1v1+…+arvr = b1v1+…+brvr
⟺
ai=bi, ∀i=1,…,r
omitted vj from list β, r>=2
v1,…,vj-hat,…,vr
* linearly indep if β is
β linearly indep. and does not span V ⟹
β, v linearly independent, ∀v∉β
β linearly indep. and span(β)=V, then c1v1+…+crvr=0 nontrivial ⟹
v1,…,vj-hat,…,vr spans V, cj≠0
BASIS of vector space V
β linearly independent: span(β)=V
A = [a1 … an] ∈ Mn(F) invertible ⟹
a1,…,an is basis for F^n
REPLACEMENT LEMMA
- β = u1,…,ur spans V≠∅
- v= Σciui ≠ 0
⟹
- ∃j: cj ≠ 0
- cj ≠ 0 ⟹ v, u1,…, uj-hat,…, ur spans V
- cj ≠ 0 and β basis for V
⟹
v, u1,…, uj-hat,…, ur is basis for V - β basis for V, r>=2
- v∉span{u1,…,uk}, k∈{1,2,…,r}
⟹
∃j∈{k+1,k+2,…,r}: v, u1,…, uk, uk+1, uj-hat,…, ur is basis for V
βr basis for V, γn linearly independent
⟹
⟹ n<=r
- n=r ⟹ γ basis for V
βr, γn bases for V⟹
n=r
DIMENSION V
- {v1,…,vn} is basis of V
⟹ dimV=n - V=∅ ⟹ dimV=0
DIMENSION MATRIX
- A = [a1 … an] ∈ F^(mxn)
- β=a1,…,an
⟹
dim span β = dim col A =: rank A
span(v1,…,vr)=V
⟹
- dimV=n<=r
- ∃i1,…,in ∈ {1,…,r}: {vi1,…,vin} basis for V
v1,…,vr linearly independent, dimV=n>r
⟹
∃w1,…,wn-r∈V: v1,…, vr, w1,…, wn-r basis for V
β = v1,…,vn, dimV=n ⟹
- β spans V ⟹ β basis for V
- β linearly independent ⟹ β basis for V
U subspace of V, dimV=n
⟹
- dimU<=n
- dimU=n ⟺ U=V
U, W subspaces of V, dimV<∞
⟹
dim(U ∩ W) + dim (U + W) = dim U + dim W
U, W subspaces of V, dimV<∞, k>0
⟹
- dimU + dimV > dim V ⟹ ∃v∈(U ∩ W): v≠0
- dimU + dimV >= dim V + k
⟹ U ∩ W contains k linearly independent vectors
A, B, C ∈ Mn(F): AB=I=BC ⟹
A=C
A, B ∈ Mn(F) ⟹
AB=I ⟺ BA=I
β-BASIS REPRESENTATION FUNCTION
- the function [.]_β :V->Fn defined by [u]_β = [c1…cn]^T
where β = v1,…,vn is basis for finite-dim. V and u∈V is any vector written as (unique) linear comb. u= c1v1+…+cnvn - c1,…,cn are “coordinates of u” w.r.t. basis β.
- [u]_β is “β-coordinate vector of u”
LINEAR TRANSFORMATION
T:V->W
T(cu+v) = cT(u) +T(v) ∀c∈F,∀u,v∈V
LINEAR OPERATOR
T:V->W
V=W
SET OF LINEAR TRANSFORMATIONS
SET OF LINEAR OPERATORS
L(V,W)
L(V)
LINEAR TRANSFORMATION INDUCED BY A
T_A: F^(n) → F^(n): x↦Ax, A∈Mmxn(F)
KerT
T:V->W
= {v∈V|T(v) = 0}
RanT
T:V->W
= {w∈W|∃v ∈V : T(v)=w}
T∈L(V,W) is one-to-one ⟺
ker(T) = {0}
LINEAR TRANSFORMATION PROPERTIES
- T(cv) = cT(v)
- T(0) = 0
- T(-v) = -T(v)
- T(a1v1+…+anvn) = a1T(v1)+…+anT(vn)
- β = v1,…, vn basis for V, T∈L(V,W)
- v = c1v1+…cnvn
⟹
- Tv = c1Tv1+…cnTvn
- RanT = span(Tv1,…,Tvn}
- dimRanT <= n
β-γ change of basis matrix
γ[I]β = [[v1]γ … [vn]γ]
(describes how to represent each vector in basis β as linear combination of vectors in the basis γ)
inverse of γ[I]β
β[I]γ = [[w1] β … [wn] β]
- β = v1,…, vn basis for V, dimV=n>0
- S∈Mn(F) invertible
⟹
∃γ basis for V: S=β[I]γ
β = a1,…, an basis for F^(n)
⟹
A = [a1…an] ∈ Mn(F) invertible
A∈Mn(F) invertible ⟺
rankA=n
- dimV=n>0
- β, γ bases for V
- S= γ[I]β
⟹
- S invertible
- γ[T]γ = S β[T]β S^(-1)
- dimV=n>0
- S invertible
- β basis for V
⟹
∃γ basis for V: γ[T]γ = S β[T]β S^(-1)
A, B∈Mn(F) “similar over F”
∃S∈Mn(F): A=SBS^(-1)
A,B∈Mn(F) similar ⟺
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 similar to B
A,B∈Mn(F) similar
⟹
- A-λI similar to B-λI, ∀λ∈F
- TrA=TrB
- detA=detB
Similarity is equivalence relation
- reflexive
- symmetric
- transitive
DIMENSION THEOREM FOR LINEAR TRANSFORMATIONS
dim ker T + dim ran T = dim V,
T∈L(V,W)
dimV=dimW, T∈L(V,W) ⟹
kerT=∅ ⟺ ranT=W
DIMENSION THEOREM FOR MATRICES
- dim null A + dim col A = n
- m=n ⟹ [nullA= ∅ ⟺colA=Fn]
A ∈Mmxn(F)
INNER PRODUCT on V
function <.,.> : VxV -> F satisfying ∀u,v,w∈V and ∀c∈F:
- <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>__
INNER PRODUCT SPACE
vector space V endowed with innerproduct
standard inner product on F^n
<u,v> = v*u = Σui(vi)_
standard inner product on Pn
<p,q> = ∫p(t)(q(t))_dt
standard inner product on Mn(F)
<A,B> = tr(B*A) = Σaij(bij)_
standard inner product on C(F, [a,b])
<f,g> = ∫(a,b)p(t)(q(t))_dt
*when[a,b] = [-π,π], divide integral by π
ORTHOGONAL u,v∈V
<u,v> = 0
u⊥v
ORTHOGONAL SUBSETS A,B ⊆V
every u∈A, v∈B
u⊥v
ORTHOGONAL PROPERTIES
- 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>
referred to as norm on V
Euclidean norm
||v||2 = √<v,v> = √(Σ|vi|^2)
Frobenius norm
||A||2 = √<A,A> = tr(A*A) = Σ|aij|^2
L^2 norm
||f|| = √( ∫(a,b)|f(t)|^2dt)
DERIVED NORM PROPERTIES
- “nonegativity”
||u|| real >=0 - “positivity”
||u||= 0 ⟺ u=0 - “homogeneity”
||cu|| = |c|||u|| - “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
UNIT VECTOR u
||u|| = 1
NORMALISATION of u
u/||u||
CAUCHY SCHWARZ INEQUALITY
- |<u,v>| <= ||u||||v||
- |<u,v>| = ||u||||v||
⟺
u,v linearly dependent
i.e. one is scalar multiple of other
TRIANGLE INEQUALITY FOR DERIVED NORM
- ||u+v||<= ||u||+||v||
- ||u+v||= ||u||+||v||
⟺
one is real non-neg. scalar multiple of other
POLARISATION IDENTITIES (4.5.24)
u,v∈V
* F=R
⟹
<u,v> = 1/4 (||u+v||^2 - ||u-v||^2)
- F=C
⟹
<u,v> = 1/4 (||u+v||^2 - ||u-v||^2 + i||u+iv||^2 - i||u-iv||^2)
NORM on V
function ||.|| : V -> [0, ∞) with following properties for ∀u,v∈V and ∀c∈F:
- ||u|| real >=0
- ||u||=0 ⟺ u=0
- ||cu|| = |c|||u||
- ||u+v||<= ||u||+||v||
l1 norm
||u||1 = |u1| + … + |un|
l∞ norm
||u||∞ = max{|ui|: 1<=i<=n}
Euclidean norm
||u||2 = √ (|u1|^2 + … + |un|^2)
UNIT BALL of normed space V
{v∈V: ||v||<=1}
