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linear algebra 2 > NOTES > Flashcards

NOTES Flashcards

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
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2
Q

V “zero vector space”

A

V={0}

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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

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4
Q

U⊆V subpace of V
⟺

A

cu+v ∈U whenever u,v ∈U and c ∈F

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5
Q

LINEAR COMBINATION of v1,…,vr ∈V

A

the expression c1v1 + … + crvr
for given c1,…,cr ∈F

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6
Q

span(v1,..,vr∈V)

A

set of all linear combinations of v1,…,vr
:= {c1v1+…+crvr|c1,…,cr∈F} ⊆V

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7
Q

v∈span(v1,…,vn)

A
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8
Q

span(∅)

A

= {0}

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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

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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

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11
Q

span(U ∪ W) when U,W subspaces of V

A

= U + W
is subspace of V

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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

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13
Q

v1,…,vr LINEARLY DEPENDENT

A

if there are scalars c1,…,cr ∈F not all zero such that
c1v1+…+crvr=0

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14
Q

v1,…,vr LINEARLY INDEPENDENT

A

not linearly dependent
⟺
c1=…=cr=0

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15
Q

Thm.1.6.11

A

v1,…,vr linearly independent
then a1v1+…+arvr = b1v1+…+brvr
⟺ ai=bi each i=1,…,r

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16
Q

BASIS of vector space V

A

v1,…,vr∈V
if linearly independent and span(v1,…,vr)=V

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17
Q

MATRIX BASIS

A

Let A = [a1 … an] ∈ Mn(F) be invertible then a1,…,an is basis for Fn

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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
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19
Q

DIMENSION V

A
  • {v1,…,vn} is basis of V
    ⟹ V dimension is n
  • V={0} ⟹ dimension 0
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20
Q

FINITE DIMENSIONAL

A
  • V has dimension n integer
    ⟹ dimension denoted dimV
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21
Q

INFINITE DIMENSIONAL

A
  • V not finite dimensional
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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

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23
Q

dim(U ∩ W) + dim (U + W) =

A

dim U + dim W

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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”
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25
Q

LINEAR TRANSFORMATION
T:V->W

A

T(cu+v) = cT(u) +T(v) ∀c∈F,∀u,v∈V

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26
Q

LINEAR OPERATOR

A

V=W

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27
Q

SET OF LINEAR TRANSFORMATIONS/OPERATORS

A

L(V,W)/L(V)

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28
Q

LINEAR TRANSFORMATION INDUCED BY A

A

T_A

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29
Q

T∈L(V,W) is one-to-one ⟺

A

ker(T) = {0}

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30
Q

LINEAR TRANSFORMATION PROPERTIES

A
  • T(cv) = cT(v)
  • T(0) = 0
  • T(-v) = -T(v)
  • T(a1v1+…+anvn) = a1T(v1)+…+anT(vn)
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31
Q

KerT
T:V->W

A

= {v∈V|T(v) = 0}

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32
Q

RanT
T:V->W

A

= {w∈W|∃v ∈V : T(v)=w}

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33
Q

SYLVESTER EQUATION

A

T(X) = AX+XB = C

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34
Q

β-γ change of basis matrix

A

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
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35
Q

inverse of γ[I]β

A

β[I]γ = [[w1] β … [wn] β]

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36
Q

cor.2.4.11,12

A
  1. if a1,…,an is basis of Fn then A=[a1…an] ∈Mn(F) is invertible
  2. A∈Mn(F) invertible ⟺ rankA=n
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37
Q

A, B∈Mn(F) SIMILAR

A

if there is invertible S∈Mn(F) such that A = SBS^(-1)

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38
Q

A,B∈Mn(F) similar ⟺

A

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
Q

A,B∈Mn(F) similar ⟹

A
  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
Q

EQUIVALENCE RELATION

A

a relation between pair of matrices that is reflexive, symmetric and transitive.

41
Q

REFLEXIVE

A

A similar to A

42
Q

SYMMETRIC

A

A similar to B
⟹
B similar to A

43
Q

TRANSITIVE

A

A similar to B and B similar to C
⟹
A similar to C

44
Q

DIMENSION THEOREM FOR LINEAR TRANSFORMATIONS

A

T∈L(V,W), V finite dim.
dim ker T + dim ran T = dim V

45
Q

DIMENSION THEOREM FOR MATRICES
A ∈Mmxn(F)

A
  • dim null A + dim col A = n
  • if m=n then nullA={0} ⟺colA=Fn
46
Q

dim COL (A) = dim ROW(A) =

A

number of leading 1s in row reduced echelon form

47
Q

NULL (A)

A

ker(A)

48
Q

INNER PRODUCT on V

A

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>__

49
Q

INNER PRODUCT SPACE

A

vector space V endowed with innerproduct

50
Q

ORTHOGONAL u,v∈V

A

<u,v> = 0
u⊥v

51
Q

ORTHOGONAL SUBSETS A,B ⊆V

A

every u∈A, v∈B
u⊥v

52
Q

ORTHOGONAL PROPERTIES

A
  • u⊥v ⟺ v⊥u
  • 0⊥u, ∀u∈V
  • v⊥u, ∀u∈V ⟹ v=0
53
Q

<u,v> = <u,w>, ∀u∈V ⟹

A

v=w

54
Q

NORM DERIVED FROM INNER PRODUCT

A

||v|| = √<v,v>

referred to as norm on V

55
Q

DERIVED NORM PROPERTIES

A
  • ||u|| real >=0
  • ||u||= 0 ⟺ u=0
  • ||cu|| = |c|||u||
  • <u,v>=0 ⟹ ||u+v||^2 = ||u||^2 + ||v||^2
  • ||u+v||^2 + ||u-v||^2
    = 2||u||^2 + 2||v||^2
56
Q

UNIT VECTOR u

A

||u|| = 1

57
Q

CAUCHY SCHWARZ INEQUALITY

A

|<u,v>| <= ||u||||v||

and equal ⟺ u,v linearly dependent i.e. one is scalar multiple of other

58
Q

TRIANGLE INEQUALITY FOR DERIVED NORM

A

||u+v||<= ||u||+||v||

and equal ⟺ one is real non-neg. scalar multiple of other

59
Q

POLARISATION IDENTITIES (4.5.24)

A

V F-inner-product space and u,v∈V
- if F=R, then
<u,v> = 1/4 (||u+v||^2 - ||u-v||^2)
- if F=C, then
<u,v> = 1/4 (||u+v||^2 - ||u-v||^2 + i||u+iv||^2 - i||u-iv||^2)

60
Q

NORM on V

A

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
Q

NORMED VECTOR SPACE

A

real or complex V with norm

62
Q

UNIT BALL of normed space V

A

{v∈V: ||v||<=1}

63
Q

UNIT VECTOR u

A

if ||u|| = 1

64
Q

NORMALISATION

A
  • u/||u||
65
Q

ORTHONORMAL VECTORS

A

u1,u2,… (finite or infinite) such that
<ui,uj> = δij for all i,j

66
Q

δij

A

= 1 if i=j
= 0 if not

67
Q

ORTHONORMAL SYSTEM

A

orthonormal sequence of vectors u1,…,un
⟹
|| Σaiui||^2 = Σ|ai|^2 for all ai ∈F
⟹
ui linearly independent

68
Q

ORTHONORMAL BASIS

A

basis for a finite-dimensional inner product space that is orthonormal system

69
Q

GRAM-SCHMIDT THEOREM

A

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
Q

GRAM SCHMIDT PROCESS

A

u1=v1
uk = vk - Σ(<vk,uj>uj)/||uj||^2
ek = uk/||uk||

sum from j=1 to k-1

71
Q

LINEAR FUNCTIONAL

A

a linear transformation φ:V->F
where V is F-vectorspace

72
Q

RIESZ REPRESENTATION THEOREM

A
  • 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,w>, ∀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
Q

RIESZ VECTOR FOR LINEAR FUNCTIONAL φ

A

w

74
Q

ADJOINT T*:W->V of T:V->W

A

if <Tv,w>_W = <v,T*(w)>_V , ∀v∈V, ∀w∈W

75
Q

SELF-ADJOINT T

A

if T=T*

76
Q

HERMITIAN A

A

if square matrix A=A*

77
Q

ORTHOGONAL COMPLEMENT

A

if U nonempty subset of inner product space V
U^⊥ = {v∈V:<u,v>=0, ∀u∈U}

  • if U=∅, then U^⊥ = V
78
Q

s MINIMUM NORM SOLUTION of Ax=y

A

if ||s||2<=||u||
2 whenever Au=y
for A(mxn) and y ∈colA

79
Q

ORTHOGONAL PROJECTION of v onto u

A

the linear operator P_U∈L(V) defined by:
(P_U)v = Σ<v,ui>ui, ∀v∈V

for finite dim. subspace U of V with u1,…,ur orthonormal basis of U
(sum from i=1 to r)

80
Q

BEST APPROXIMATION THEOREM

A

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
Q

NORMAL EQUATIONS

A

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 [[<u1,ui>]…[<un,ui>]][ci] = [<v,ui>] ()
- the system () is consistent
- if u1,…,un linearly independent then (*) has unique solution

82
Q

GRAM MATRIX of u1,..,un

A

G(u1,…,un) = [[<u1,ui>]…[<un,ui>]]
if u1,…,un vectors in inner product space

83
Q

GRAM DETERMINANT of u1,…,un

A

g(u1,…,un) = detG(u1,..,un)

84
Q

PROPERTIES OF GRAM MATRIX

A
  • hermitian
  • positive semi-definite
85
Q

LEAST SQUARE SOLUTION OF AN INCONSISTENT LINEAR SYSTEM

A
  • if A∈M(mxn) and y∈Fm then x0∈Fn satisfies
    min(x∈Fn) ||y-Ax||_2 = ||y-Ax0||_2
    ⟺
    AAx0 = Ay
  • the system is always consistent, it has a unique solution if rankA=n
86
Q

A∈Mn(F) POSITIVE SEMI-DEFINITE

A

A hermitian and <Ax,x> >= 0 for all x∈Fn

87
Q

A∈Mn(F) POSITIVE DEFINITE

A

A hermitian and <Ax,x> > 0 for all nonzero x∈Fn

88
Q

A∈Mn(F) NEGATIVE SEMI-DEFINITE

A

if -A is positive semidefinite

89
Q

A∈Mn(F) NEGATIVE DEFINITE

A

if -A is positive definite

90
Q

SINGULAR VALUES

A
  • δ1,…, δr where δ1>=…>= δr>0 and (δi)^2= λi
    where λi is eigenvalue of AA or AA
91
Q

SINGULAR VALUE DECOMPOSITION

A
  • 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
Q

MULTIPLICITY OF δi

A

= 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
Q

A IDEMPOTENT

A

A^2=A

94
Q

A ORTHOGONAL PROJECTION

A

A is idempotent and symmetric
ranA=colA
ker(A)=ran(A)^ ⊥

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