NECESSARY Flashcards
U⊆V subpace of V
⟺
U nonempty and cu+v ∈U whenever u,v ∈U and c ∈F
span(v1,..,vr∈V)
set of all linear combinations of v1,…,vr
:= {c1v1+…+crvr|c1,…,cr∈F} ⊆V
spanning set of V
{v1,..,vr∈V} if every vector in V can be written as linear comb. of v1,..,vr
span(v1,..,vr) = V
span(U ∪ W) when U,W subspaces of V
= U + W
is subspace of V
DIRECT SUM
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
BASIS of vector space V
v1,…,vr∈V
if linearly independent and span(v1,…,vr)=V
MATRIX BASIS
Let A = [a1 … an] ∈ Mn(F) be invertible then a1,…,an is basis for Fn
DIMENSION MATRIX
Let A = [a1 … an] ∈ Mmxn(F) and β=a1,…,an in Fm.
then dim span β = dim col A = rank A
dim(U ∩ W) + dim (U + W) =
dim U + dim W
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 INDUCED BY A
T_A
T∈L(V,W) is one-to-one ⟺
ker(T) = {0}
KerT
T:V->W
= {v∈V|T(v) = 0}
RanT
T:V->W
= {w∈W|∃v ∈V : T(v)=w}
SYLVESTER EQUATION
T(X) = AX+XB = C
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
inverse of γ[I]β
β[I]γ = [[w1] β … [wn] β]
cor.2.4.11,12
- if a1,…,an is basis of Fn then A=[a1…an] ∈Mn(F) is invertible
- A∈Mn(F) invertible ⟺ rankA=n
A, B∈Mn(F) SIMILAR
if there is invertible S∈Mn(F) such that A = SBS^(-1)
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
A,B∈Mn(F) similar ⟹
- 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) - TrA=TrB and detA=detB
EQUIVALENCE RELATION
a relation between pair of matrices that is:
1. reflexive,
(A similar to A)
- symmetric,
(A similar to B
⟹
B similar to A) - transitive,
(A similar to B and B similar to C
⟹
A similar to C)
DIMENSION THEOREM FOR LINEAR TRANSFORMATIONS
T∈L(V,W), V finite dim.
dim ker T + dim ran T = dim V
DIMENSION THEOREM FOR MATRICES
A ∈Mmxn(F)
- dim null A + dim col A = n
- if m=n then nullA={0} ⟺colA=Fn
dim COL (A) = dim ROW(A) =
number of leading 1s in row reduced echelon form
NULL (A)
ker(A)
ORTHOGONAL u,v∈V
<u,v> = 0
u⊥v
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>__
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
DERIVED NORM PROPERTIES
- ||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
CAUCHY SCHWARZ INEQUALITY
|<u,v>| <= ||u||||v||
and equal ⟺ u,v linearly dependent i.e. one is scalar multiple of other
TRIANGLE INEQUALITY FOR DERIVED NORM
||u+v||<= ||u||+||v||
and equal ⟺ one is real non-neg. scalar multiple of other
POLARISATION IDENTITIES (4.5.24)
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)
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||
NORMALISATION
- u/||u||
ORTHONORMAL VECTORS
u1,u2,… (finite or infinite) such that
<ui,uj> = δij for all i,j
δij
= 1 if i=j
= 0 if not
ORTHONORMAL SYSTEM
orthonormal sequence of vectors u1,…,un
⟹
|| Σaiui||^2 = Σ|ai|^2 for all ai ∈F
⟹
ui linearly independent
ORTHONORMAL BASIS
basis for a finite-dimensional inner product space that is orthonormal system
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
GRAM SCHMIDT PROCESS
u1=v1
uk = vk - Σ(<vk,uj>uj)/||uj||^2
ek = uk/||uk||
sum from j=1 to k-1
LINEAR FUNCTIONAL
a linear transformation φ:V->F
where V is F-vectorspace
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,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
RIESZ VECTOR FOR LINEAR FUNCTIONAL φ
w
HERMITIAN A
if square matrix A=A*
ORTHOGONAL COMPLEMENT
if U nonempty subset of inner product space V
U^⊥ = {v∈V:<u,v>=0, ∀u∈U}
- if U=∅, then U^⊥ = V
s MINIMUM NORM SOLUTION of Ax=y
if ||s||2<=||u||
2 whenever Au=y
for A(mxn) and y ∈colA
ORTHOGONAL PROJECTION of v onto u
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)
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
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 [[<u1,ui>]…[<un,ui>]][ci] = [<v,ui>] ()
- the system () is consistent
- if u1,…,un linearly independent then (*) has unique solution
GRAM MATRIX of u1,..,un
G(u1,…,un) = [[<u1,ui>]…[<un,ui>]]
if u1,…,un vectors in inner product space
best approximation of v w.r.t. V
projection of v onto V
PROPERTIES OF GRAM MATRIX
- hermitian
- positive semi-definite
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
⟺
AAx0 = Ay - the system is always consistent, it has a unique solution if rankA=n
A∈Mn(F) POSITIVE SEMI-DEFINITE
A hermitian and <Ax,x> >= 0 for all x∈Fn
A∈Mn(F) POSITIVE DEFINITE
A hermitian and <Ax,x> > 0 for all nonzero x∈Fn
A∈Mn(F) NEGATIVE SEMI-DEFINITE
if -A is positive semidefinite
A∈Mn(F) NEGATIVE DEFINITE
if -A is positive definite
SYMMETRIC PROPERTIES
- Has real eigenvalues
- Eigenvectors corresponding to the eigenvalues are orthogonal
SINGULAR VALUES
- δ1,…, δr where δ1>=…>= δr>0 and (δi)^2= λi
where λi is eigenvalue of AA or AA
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”
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”
A IDEMPOTENT
A^2=A
A ORTHOGONAL PROJECTION
A is idempotent and symmetric
ranA=colA
ker(A)=ran(A)^ ⊥
