Algebra II - Linear Algebra

Exam revision

Vector Space (6)

1. commutativity 2. associativity 3. additive identity 4. additive inverse 5. multiplicative identity 6. distributive

Subspace (3)

1. additive identity 2. closed under addition 3. closed under scalar multiplication

dim(V+U)=

dimV + dimU -dim(V and U)

Linear Map T: V→W (2)

1. additivity 2. homogeneity

T: V→W nullT=

{v in V| Tv=0}

T: V→W rangeT=

{Tv | v in V}

Rank theorem: dimV=

dim nullT + dim rangeT

Linear map is ___ and ___ <-> invertible

injective and surjective

T: V→W and S: W→V. ST=e_? and TS=e_? then S=T^-1

T: V→W and S: W→V. ST=e_V and TS=e_W then S=T^-1

Change of basis …

A linear functional is a …

A linear functional is an element of L(V, F), a linear map that sends V to a scaler.

let φ_j(v_i)=____ therefore v=__________

let φ_j(v_i)={1 if j=i and 0 if j≠i therefore v= φ₁(v)v₁+…+φ_n(v)v_n

norm. ||v||=____

||v||=sqrt(<v,v>)

given an othonormal basis e₁,..,e_n , v=______

v=<v,e₁>e₁+…+<v,e_n>e_n

Adjoint. if T: V→W , T*:___

Adjoint. if T: V→W , T*: W→V

Adjoint. if T: V→W , T*: W→V. <__,__> = <v,T*w>

<Tv,w> = <v,T*w>

T* is T ______ ______

T* is T conjugate transpose

For normal operators TT*=__ and ||Tv||=__

For normal operators TT*=T*T and ||Tv||=||T*v||

Define self-adjoint

T*=T

Positive Operators (2)

1. self adjoint 2. <Tv,v> ≥ 0

S in L(V,W). S is an isometry when ____

S is an isometry when ||Sv||=||v||

S in L(V,W). S is an unitary when ____ ____

S is an unitary when S is an invertible isometry

The eigenvalues of an unitary are ___

The eigenvalues of an unitary are |1|

A can be QR factorised if it has ____

A can be QR factorised if it has linearly independent columns.

QR factorisation. how to find Q?

Q= gram-schmidt the columns of A

QR factorisation. how to find R?

SVD. the singular values of T. s_i=

SVD. the singular values of T. s_i= sqrt(eigenvalues of T*T)

SVD. e_i=

SVD. e_i= the normalised eigenspaces of T*T

SVD. f_i=

SVD. f_i= 1/s_i Te_i

SVD. Tv=_____+…+_____

SVD. Tv= s₁<v,e₁>f₁+…+s_n<v,e_n>f_n

Matrix SVD: A = M(T) = ___

Matrix SVD: A = M(T) = FSE*

Inner Product (5)

1. positivity 2. definiteness 3. additivity in first slot 4. homogeneity in first slot 5. conjugate symmetry