NECESSARY

Question 1

U⊆V subpace of V
⟺

Answer 1

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

Question 2

span(v1,..,vr∈V)

Answer 2

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

Question 3

spanning set of V

Answer 3

{v1,..,vr∈V} if every vector in V can be written as linear comb. of v1,..,vr
span(v1,..,vr) = V

Question 4

span(U ∪ W) when U,W subspaces of V

Answer 4

= U + W
is subspace of V

Question 5

DIRECT SUM

Answer 5

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

Question 6

BASIS of vector space V

Answer 6

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

Question 7

MATRIX BASIS

Answer 7

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

Question 8

DIMENSION MATRIX

Answer 8

Let A = [a1 … an] ∈ Mmxn(F) and β=a1,…,an in Fm.
then dim span β = dim col A = rank A

Question 9

dim(U ∩ W) + dim (U + W) =

Answer 9

dim U + dim W

Question 10

β-BASIS REPRESENTATION FUNCTION

Answer 10

• 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”

Question 11

LINEAR TRANSFORMATION INDUCED BY A

Answer 11

T_A

Question 12

T∈L(V,W) is one-to-one ⟺

Answer 12

ker(T) = {0}

Question 13

KerT
T:V->W

Answer 13

= {v∈V|T(v) = 0}

Question 14

RanT
T:V->W

Answer 14

= {w∈W|∃v ∈V : T(v)=w}

Question 15

SYLVESTER EQUATION

Answer 15

T(X) = AX+XB = C

Question 16

β-γ change of basis matrix

Answer 16

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

Question 17

inverse of γ[I]β

Answer 17

β[I]γ = [[w1] β … [wn] β]

Question 18

cor.2.4.11,12

Answer 18

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

Question 19

A, B∈Mn(F) SIMILAR

Answer 19

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

Question 20

A,B∈Mn(F) similar ⟺

Answer 20

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

Question 21

A,B∈Mn(F) similar ⟹

Answer 21

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

Question 22

EQUIVALENCE RELATION

Answer 22

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

2. symmetric,
   (A similar to B
   ⟹
   B similar to A)
3. transitive,
   (A similar to B and B similar to C
   ⟹
   A similar to C)

Question 23

DIMENSION THEOREM FOR LINEAR TRANSFORMATIONS

Answer 23

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

Question 24

DIMENSION THEOREM FOR MATRICES
A ∈Mmxn(F)

Answer 24

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

Question 25

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

Answer 25

number of leading 1s in row reduced echelon form

Question 26

NULL (A)

Answer 26

ker(A)

Question 27

ORTHOGONAL u,v∈V

Answer 27

= 0 u⊥v

Question 27

INNER PRODUCT on V

Answer 27

function <.,.> : VxV -> F satisfying ∀u,v,w∈V and ∀c∈F: - real >=0 - = 0 ⟺ v=0 - = + - = c - = __

Question 28

ORTHOGONAL SUBSETS A,B ⊆V

Answer 28

every u∈A, v∈B u⊥v

Question 29

ORTHOGONAL PROPERTIES

Answer 29

- u⊥v ⟺ v⊥u - 0⊥u, ∀u∈V - v⊥u, ∀u∈V ⟹ v=0

Question 30

= , ∀u∈V ⟹

Answer 30

v=w

Question 31

NORM DERIVED FROM INNER PRODUCT

Answer 31

||v|| = √ referred to as norm on V

Question 32

DERIVED NORM PROPERTIES

Answer 32

- ||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

Question 33

CAUCHY SCHWARZ INEQUALITY

Answer 33

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

Question 34

TRIANGLE INEQUALITY FOR DERIVED NORM

Answer 34

||u+v||<= ||u||+||v|| and equal ⟺ one is real non-neg. scalar multiple of other

Question 35

POLARISATION IDENTITIES (4.5.24)

Answer 35

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)

Question 36

NORM on V

Answer 36

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||

Question 37

NORMALISATION

Answer 37

- u/||u||

Question 38

ORTHONORMAL VECTORS

Answer 38

u1,u2,... (finite or infinite) such that = δij for all i,j

Question 39

δij

Answer 39

= 1 if i=j = 0 if not

Question 40

ORTHONORMAL SYSTEM

Answer 40

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

Question 41

ORTHONORMAL BASIS

Answer 41

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

Question 42

GRAM-SCHMIDT THEOREM

Answer 42

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

Question 43

GRAM SCHMIDT PROCESS

Answer 43

u1=v1 uk = vk - Σ(uj)/||uj||^2 ek = uk/||uk|| sum from j=1 to k-1

Question 44

LINEAR FUNCTIONAL

Answer 44

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

Question 45

RIESZ REPRESENTATION THEOREM

Answer 45

- 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

Question 46

RIESZ VECTOR FOR LINEAR FUNCTIONAL φ

Answer 46

w

Question 47

HERMITIAN A

Answer 47

if square matrix A=A*

Question 48

ORTHOGONAL COMPLEMENT

Answer 48

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

Question 49

s MINIMUM NORM SOLUTION of Ax=y

Answer 49

if ||s||_2<=||u||_ 2 whenever Au=y for A(mxn) and y ∈colA

Question 50

ORTHOGONAL PROJECTION of v onto u

Answer 50

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)

Question 51

BEST APPROXIMATION THEOREM

Answer 51

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

Question 52

NORMAL EQUATIONS

Answer 52

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

Question 53

GRAM MATRIX of u1,..,un

Answer 53

G(u1,...,un) = [[]...[]] if u1,...,un vectors in inner product space

Question 53

best approximation of v w.r.t. V

Answer 53

projection of v onto V

Question 54

PROPERTIES OF GRAM MATRIX

Answer 54

- hermitian - positive semi-definite

Question 55

LEAST SQUARE SOLUTION OF AN INCONSISTENT LINEAR SYSTEM

Answer 55

- 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

Question 56

A∈Mn(F) POSITIVE SEMI-DEFINITE

Answer 56

A hermitian and >= 0 for all x∈Fn

Question 57

A∈Mn(F) POSITIVE DEFINITE

Answer 57

A hermitian and > 0 for all nonzero x∈Fn

Question 58

A∈Mn(F) NEGATIVE SEMI-DEFINITE

Answer 58

if -A is positive semidefinite

Question 59

A∈Mn(F) NEGATIVE DEFINITE

Answer 59

if -A is positive definite

Question 60

SYMMETRIC PROPERTIES

Answer 60

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

Question 61

SINGULAR VALUES

Answer 61

- δ1,..., δr where δ1>=...>= δr>0 and (δi)^2= λi where λi is eigenvalue of A*A or AA*

Question 62

SINGULAR VALUE DECOMPOSITION

Answer 62

- 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"

Question 63

MULTIPLICITY OF δi

Answer 63

= 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"

Question 64

A IDEMPOTENT

Answer 64

A^2=A

Question 65

A ORTHOGONAL PROJECTION

Answer 65

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