NOTES Flashcards
(94 cards)
1
Q
VECTOR SPACE
A
- ∃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
2
Q
V “zero vector space”
A
V={0}
3
Q
U is subspace of V (def.)
A
if U is subset of V that is a vector space with same vector addition and scalar multiplication as in V
4
Q
U⊆V subpace of V
⟺
A
cu+v ∈U whenever u,v ∈U and c ∈F
5
Q
LINEAR COMBINATION of v1,…,vr ∈V
A
the expression c1v1 + … + crvr
for given c1,…,cr ∈F
6
Q
span(v1,..,vr∈V)
A
set of all linear combinations of v1,…,vr
:= {c1v1+…+crvr|c1,…,cr∈F} ⊆V
7
Q
v∈span(v1,…,vn)
A
8
Q
span(∅)
A
= {0}
9
Q
PROPERTIES OF SPAN (thm.1.4.9,10)
A
let U,W ⊆ V
1. span U subspace of V
2. U ⊆ span U
3. U = span U ⟺ U is subspace of V
4. span(spanU) = span U
5. U ⊆W ⟹ span U ⊆ span W
10
Q
spanning set of V
A
{v1,..,vr∈V} if every vector in V can be written as linear comb. of v1,..,vr
span(v1,..,vr) = V
11
Q
span(U ∪ W) when U,W subspaces of V
A
= U + W
is subspace of V
12
Q
DIRECT SUM
A
whenever U,W subspaces of V and U ∩ W = {0}
i.e. each vector in U+W can be expressed as unique sum of vector in U and vector in W
13
Q
v1,…,vr LINEARLY DEPENDENT
A
if there are scalars c1,…,cr ∈F not all zero such that
c1v1+…+crvr=0
14
Q
v1,…,vr LINEARLY INDEPENDENT
A
not linearly dependent
⟺
c1=…=cr=0
15
Q
Thm.1.6.11
A
v1,…,vr linearly independent
then a1v1+…+arvr = b1v1+…+brvr
⟺ ai=bi each i=1,…,r
16
Q
BASIS of vector space V
A
v1,…,vr∈V
if linearly independent and span(v1,…,vr)=V
17
Q
MATRIX BASIS
A
Let A = [a1 … an] ∈ Mn(F) be invertible then a1,…,an is basis for Fn
18
Q
REPLACEMENT LEMMA
A
- suppose β=u1,…,ur spans non-zero vector space V
- let nonzero v∈V be v= Σciui from i=1 to r
THEN - cj ≠ 0 for some j=1,…,r
- cj ≠ 0 ⟹ v,u1,…,u(^)j,…,ur (*) spans V
- β basis for V and cj ≠ 0 ⟹ (*) is basis for V
- r>=2, β basis for V and v∉span{u1,…,uk} for some k ∈{k+1,k+2,…,r} such that v,u1,…,uk,uk+1,…,u(^)j,…,ur is a basis for V
19
Q
DIMENSION V
A
- {v1,…,vn} is basis of V
⟹ V dimension is n - V={0} ⟹ dimension 0
20
Q
FINITE DIMENSIONAL
A
- V has dimension n integer
⟹ dimension denoted dimV
21
Q
INFINITE DIMENSIONAL
A
- V not finite dimensional
22
Q
DIMENSION MATRIX
A
Let A = [a1 … an] ∈ Mmxn(F) and β=a1,…,an in Fm.
then dim span β = dim col A = rank A
23
Q
dim(U ∩ W) + dim (U + W) =
A
dim U + dim W
24
Q
β-BASIS REPRESENTATION FUNCTION
A
- 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”
25
LINEAR TRANSFORMATION
T:V->W
T(cu+v) = cT(u) +T(v) ∀c∈F,∀u,v∈V
26
LINEAR OPERATOR
V=W
27
SET OF LINEAR TRANSFORMATIONS/OPERATORS
L(V,W)/L(V)
28
LINEAR TRANSFORMATION INDUCED BY A
T_A
29
T∈L(V,W) is one-to-one ⟺
ker(T) = {0}
30
LINEAR TRANSFORMATION PROPERTIES
- T(cv) = cT(v)
- T(0) = 0
- T(-v) = -T(v)
- T(a1v1+...+anvn) = a1T(v1)+...+anT(vn)
31
KerT
T:V->W
= {v∈V|T(v) = 0}
32
RanT
T:V->W
= {w∈W|∃v ∈V : T(v)=w}
33
SYLVESTER EQUATION
T(X) = AX+XB = C
34
β-γ change of basis matrix
γ[I]β = [[v1]γ ... [vn]γ]
β = {v1,...,vn}
γ = {w1,...,wn} bases for V
* describes how to represent each vector in basis β as linear combination of vectors in basis γ
35
inverse of γ[I]β
β[I]γ = [[w1] β ... [wn] β]
36
cor.2.4.11,12
1. if a1,...,an is basis of Fn then A=[a1...an] ∈Mn(F) is invertible
2. A∈Mn(F) invertible ⟺ rankA=n
37
A, B∈Mn(F) SIMILAR
if there is invertible S∈Mn(F) such that A = SBS^(-1)
38
A,B∈Mn(F) similar ⟺
there is n-dim. V, with bases β and γ and linear operator T ∈L(V) such that A=β[T]β and B= γ[T]γ
39
A,B∈Mn(F) similar ⟹
1. A-λI is similar to B-λI for every λ∈F
(if there is λ∈F such that (A-λI) is similar to (B-λI) then A similar to B)
2. TrA=TrB and detA=detB
40
EQUIVALENCE RELATION
a relation between pair of matrices that is reflexive, symmetric and transitive.
41
REFLEXIVE
A similar to A
42
SYMMETRIC
A similar to B
⟹
B similar to A
43
TRANSITIVE
A similar to B and B similar to C
⟹
A similar to C
44
DIMENSION THEOREM FOR LINEAR TRANSFORMATIONS
T∈L(V,W), V finite dim.
dim ker T + dim ran T = dim V
45
DIMENSION THEOREM FOR MATRICES
A ∈Mmxn(F)
- dim null A + dim col A = n
- if m=n then nullA={0} ⟺colA=Fn
46
dim COL (A) = dim ROW(A) =
number of leading 1s in row reduced echelon form
47
NULL (A)
ker(A)
48
INNER PRODUCT on V
function <.,.> : VxV -> F satisfying ∀u,v,w∈V and ∀c∈F:
- real >=0
- = 0 ⟺ v=0
- = +
- = c
- = __
49
INNER PRODUCT SPACE
vector space V endowed with innerproduct
50
ORTHOGONAL u,v∈V
51
ORTHOGONAL SUBSETS A,B ⊆V
every u∈A, v∈B
u⊥v
52
ORTHOGONAL PROPERTIES
- u⊥v ⟺ v⊥u
- 0⊥u, ∀u∈V
- v⊥u, ∀u∈V ⟹ v=0
53
v=w
54
NORM DERIVED FROM INNER PRODUCT
||v|| = √
referred to as norm on V
55
DERIVED NORM PROPERTIES
- ||u|| real >=0
- ||u||= 0 ⟺ u=0
- ||cu|| = |c|||u||
- =0 ⟹ ||u+v||^2 = ||u||^2 + ||v||^2
- ||u+v||^2 + ||u-v||^2
= 2||u||^2 + 2||v||^2
56
UNIT VECTOR u
||u|| = 1
57
CAUCHY SCHWARZ INEQUALITY
|| <= ||u||||v||
and equal ⟺ u,v linearly dependent i.e. one is scalar multiple of other
58
TRIANGLE INEQUALITY FOR DERIVED NORM
||u+v||<= ||u||+||v||
and equal ⟺ one is real non-neg. scalar multiple of other
59
POLARISATION IDENTITIES (4.5.24)
V F-inner-product space and u,v∈V
- if F=R, then
= 1/4 (||u+v||^2 - ||u-v||^2)
- if F=C, then
= 1/4 (||u+v||^2 - ||u-v||^2 + i||u+iv||^2 - i||u-iv||^2)
60
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||
61
NORMED VECTOR SPACE
real or complex V with norm
62
UNIT BALL of normed space V
{v∈V: ||v||<=1}
63
UNIT VECTOR u
if ||u|| = 1
64
NORMALISATION
- u/||u||
65
ORTHONORMAL VECTORS
u1,u2,... (finite or infinite) such that
= δij for all i,j
66
δij
= 1 if i=j
= 0 if not
67
ORTHONORMAL SYSTEM
orthonormal sequence of vectors u1,...,un
⟹
|| Σaiui||^2 = Σ|ai|^2 for all ai ∈F
⟹
ui linearly independent
68
ORTHONORMAL BASIS
basis for a finite-dimensional inner product space that is orthonormal system
69
GRAM-SCHMIDT THEOREM
V inner product space
v1,...,vn∈V linearly independent
There is orthonormal system u1,...,un such that span{v1,...,vk} = span {u1,...,uk}, k=1,...,n
70
GRAM SCHMIDT PROCESS
u1=v1
uk = vk - Σ(uj)/||uj||^2
ek = uk/||uk||
sum from j=1 to k-1
71
LINEAR FUNCTIONAL
a linear transformation φ:V->F
where V is F-vectorspace
72
RIESZ REPRESENTATION THEOREM
- let V be finite dim. F-inner product space and φ:V->F be a linear functional
1. there is a unique w∈V such that φ(v) = , ∀v∈V
2. let u1,...,un be an orthonormal basis of V. the vector w in (1) is
w= (φ(u1))(_)*u1 + (φ(un))(_)*un
73
RIESZ VECTOR FOR LINEAR FUNCTIONAL φ
w
74
ADJOINT T*:W->V of T:V->W
if _W = _V , ∀v∈V, ∀w∈W
75
SELF-ADJOINT T
if T=T*
76
HERMITIAN A
if square matrix A=A*
77
ORTHOGONAL COMPLEMENT
if U nonempty subset of inner product space V
U^⊥ = {v∈V:=0, ∀u∈U}
* if U=∅, then U^⊥ = V
78
s MINIMUM NORM SOLUTION of Ax=y
if ||s||_2<=||u||_
2 whenever Au=y
for A(mxn) and y ∈colA
79
ORTHOGONAL PROJECTION of v onto u
the linear operator P_U∈L(V) defined by:
(P_U)v = Σui, ∀v∈V
for finite dim. subspace U of V with u1,...,ur orthonormal basis of U
(sum from i=1 to r)
80
BEST APPROXIMATION THEOREM
let U be finite dim. subspace of inner prod. space V and let P_U be the orthogonal proj. onto U.
||v-(P_U)v||<=||v-u|| ∀v ∈V, ∀u ∈U
with equality ⟺u=(P_U)V
81
NORMAL EQUATIONS
U finite dim. subspace of inner product space V
span{u1,..,un}=U
- the projection of V onto U P_U V= Σcjuj where [c1 ... cn]^T is solution of the normal equations [[]...[]][ci] = [] (*)
- the system (*) is consistent
- if u1,...,un linearly independent then (*) has unique solution
82
GRAM MATRIX of u1,..,un
G(u1,...,un) = [[]...[]]
if u1,...,un vectors in inner product space
83
GRAM DETERMINANT of u1,...,un
g(u1,...,un) = detG(u1,..,un)
84
PROPERTIES OF GRAM MATRIX
- hermitian
- positive semi-definite
85
LEAST SQUARE SOLUTION OF AN INCONSISTENT LINEAR SYSTEM
- if A∈M(mxn) and y∈Fm then x0∈Fn satisfies
min(x∈Fn) ||y-Ax||_2 = ||y-Ax0||_2
⟺
A*Ax0 = A*y
- the system is always consistent, it has a unique solution if rankA=n
86
A∈Mn(F) POSITIVE SEMI-DEFINITE
A hermitian and >= 0 for all x∈Fn
87
A∈Mn(F) POSITIVE DEFINITE
A hermitian and > 0 for all nonzero x∈Fn
88
A∈Mn(F) NEGATIVE SEMI-DEFINITE
if -A is positive semidefinite
89
A∈Mn(F) NEGATIVE DEFINITE
if -A is positive definite
90
SINGULAR VALUES
- δ1,..., δr where δ1>=...>= δr>0 and (δi)^2= λi
where λi is eigenvalue of A*A or AA*
91
SINGULAR VALUE DECOMPOSITION
- define Σr = [[δ1 0 ... 0][0 δ2 0...0]...[0...0 δr]] ∈ Mr(R)
=: zero matrix with ascending singular values as diagonal
* r=rankA
- then there are unitary matrices V∈Mm(F) and W∈Mn(F) such that A=VΣW*
where
Σ = [[Σr]0] ∈ Mmxn(R) is the same size as A
if m=n, then V,W∈Mn(F) and Σ = Σr + 0n-r
* columns of V are "left singular vectors of A"
* columns of W are "right-singular vectors of A"
92
MULTIPLICITY OF δi
= the multiplicity of δ^2 as eigenvalue of A*A
* if δ zero then multiplicity:=min{m,n}-r
* if singular value has multiplicity 1 then called "simple"
* if every singular value of A is simple then they are "distinct"
93
A IDEMPOTENT
A^2=A
94
A ORTHOGONAL PROJECTION
A is idempotent and symmetric
ranA=colA
ker(A)=ran(A)^ ⊥
