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linear algebra 2 > NECESSARY > Flashcards

NECESSARY Flashcards

1
Q

U⊆V subpace of V
⟺

A

U nonempty and cu+v ∈U whenever u,v ∈U and c ∈F

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2
Q

span(v1,..,vr∈V)

A

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

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3
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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4
Q

span(U ∪ W) when U,W subspaces of V

A

= U + W
is subspace of V

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5
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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6
Q

BASIS of vector space V

A

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

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7
Q

MATRIX BASIS

A

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

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8
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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9
Q

dim(U ∩ W) + dim (U + W) =

A

dim U + dim W

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10
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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11
Q

LINEAR TRANSFORMATION INDUCED BY A

A

T_A

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12
Q

T∈L(V,W) is one-to-one ⟺

A

ker(T) = {0}

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13
Q

KerT
T:V->W

A

= {v∈V|T(v) = 0}

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14
Q

RanT
T:V->W

A

= {w∈W|∃v ∈V : T(v)=w}

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15
Q

SYLVESTER EQUATION

A

T(X) = AX+XB = C

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16
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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17
Q

inverse of γ[I]β

A

β[I]γ = [[w1] β … [wn] β]

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18
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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19
Q

A, B∈Mn(F) SIMILAR

A

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

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20
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

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21
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
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22
Q

EQUIVALENCE RELATION

A

a relation between pair of matrices that is:
1. reflexive,
(A similar to A)

  1. symmetric,
    (A similar to B
    ⟹
    B similar to A)
  2. transitive,
    (A similar to B and B similar to C
    ⟹
    A similar to C)
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23
Q

DIMENSION THEOREM FOR LINEAR TRANSFORMATIONS

A

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

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24
Q

DIMENSION THEOREM FOR MATRICES
A ∈Mmxn(F)

A
  • dim null A + dim col A = n
  • if m=n then nullA={0} ⟺colA=Fn
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25
Q

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

A

number of leading 1s in row reduced echelon form

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26
Q

NULL (A)

A

ker(A)

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27
Q

ORTHOGONAL u,v∈V

A

<u,v> = 0
u⊥v

27
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>__

28
Q

ORTHOGONAL SUBSETS A,B ⊆V

A

every u∈A, v∈B
u⊥v

29
Q

ORTHOGONAL PROPERTIES

A
  • u⊥v ⟺ v⊥u
  • 0⊥u, ∀u∈V
  • v⊥u, ∀u∈V ⟹ v=0
30
Q

<u,v> = <u,w>, ∀u∈V ⟹

A

v=w

31
Q

NORM DERIVED FROM INNER PRODUCT

A

||v|| = √<v,v>

referred to as norm on V

32
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
33
Q

CAUCHY SCHWARZ INEQUALITY

A

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

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

34
Q

TRIANGLE INEQUALITY FOR DERIVED NORM

A

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

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

35
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)

36
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||

37
Q

NORMALISATION

A
  • u/||u||
38
Q

ORTHONORMAL VECTORS

A

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

39
Q

δij

A

= 1 if i=j
= 0 if not

40
Q

ORTHONORMAL SYSTEM

A

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

41
Q

ORTHONORMAL BASIS

A

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

42
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

43
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

44
Q

LINEAR FUNCTIONAL

A

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

45
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
46
Q

RIESZ VECTOR FOR LINEAR FUNCTIONAL φ

A

w

47
Q

HERMITIAN A

A

if square matrix A=A*

48
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
49
Q

s MINIMUM NORM SOLUTION of Ax=y

A

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

50
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)

51
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

52
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

53
Q

GRAM MATRIX of u1,..,un

A

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

53
Q

best approximation of v w.r.t. V

A

projection of v onto V

54
Q

PROPERTIES OF GRAM MATRIX

A
  • hermitian
  • positive semi-definite
55
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
56
Q

A∈Mn(F) POSITIVE SEMI-DEFINITE

A

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

57
Q

A∈Mn(F) POSITIVE DEFINITE

A

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

58
Q

A∈Mn(F) NEGATIVE SEMI-DEFINITE

A

if -A is positive semidefinite

59
Q

A∈Mn(F) NEGATIVE DEFINITE

A

if -A is positive definite

60
Q

SYMMETRIC PROPERTIES

A
  1. Has real eigenvalues
  2. Eigenvectors corresponding to the eigenvalues are orthogonal
61
Q

SINGULAR VALUES

A
  • δ1,…, δr where δ1>=…>= δr>0 and (δi)^2= λi
    where λi is eigenvalue of AA or AA
62
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”
63
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”

64
Q

A IDEMPOTENT

A

A^2=A

65
Q

A ORTHOGONAL PROJECTION

A

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

linear algebra 2 (3 decks)

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