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## Filename: diary
## Description: Diary of MA20216 lectures 2018
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## Author: Francis Burstall
## Modified at: Wed Dec 12 13:50:44 2018
## Modified by: Francis Burstall
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Lecture 1: Introduction, administration and propaganda.
Chapter 1. Linear algebra: concepts and examples: defn of
vector space. Examples, including V^I.
[notes: middle of page 2]
Lecture 2: Familiar cases of F^I. Vector subspace: defn,
efficient characterisation and examples. Linearly
indep lists and spanning lists of vectors; bases; dimension.
Standard basis of F^I, examples.
[notes: end of section 1.3.1]
Lecture 3: Useful facts: extension property of linearly
indep lists; dim of subspace. Linear maps: defn,
kernel, image, isomorphism. Examples from analysis. Set of
linear maps is vector space under pointwise addition/scalar
multiplication. The matrix of a linear map with respect to
bases on domain and codomain.
[notes: top of page 6]
Lecture 4: Fancy explanation of linear maps vs matrices,
bases give linear isomorphism F^n to V. Extension by
linearity. Rank-nullity: statement. Application: if dim
V=dim W, a linear map from V to W injects if and only if
surjects if and only if bijects.
Chapter 2: Sums and quotients. Sum of subspaces:
defn and characterisation as minimal subspace containing the
summands (statement).
[notes: page 9: middle]
Lecture 5: Characterisation of sums: proof. Direct sums.
Criterion for a sum to be direct. Two summand case:
complements. Projections: defn, any two summand direct sum
comes from projections---statement and first half of proof.
[notes: top of page 12]
Lecture 6: Finish proof of projections from direct
sums. Corollary: dimensions add for two summand direct sums.
Inductively defining direct sums. Application: dimensions
add for arbitrary direct sums. Sum is direct iff bases of
summands concatenate to a basis of the sum. Complements
exist for subspaces of finite-diml V. Extension from a
subspace: statement.
[notes: middle of page 14]
Lecture 7: Extension from a subspace: proof. Quotients:
congruence modulo a subspace U; this is an equivalence
relation; equivalence classes are cosets; example: fibres
of linear maps; quotient space V/U is set of cosets of U. V/U is
a vector space for which quotient map q:V->V/U is linear
surjection with kernel U (statement).
[notes: statement of (2.13), near top of page 16]
Lecture 8: V/U is a vector space for which quotient map
q:V->V/U is linear surjection with kernel U (proof). Dim
V/U = dim V - dim U. How to think about quotients: V/U is
vector space with surjective linear map q:V -> V/U with
kernel U: this is all you need to know. First Isomorphism
Theorem.
[notes: end of Chapter 2]
Lecture 9: New chapter: inner product spaces. Revise dot
product in R^n and C^n. Defn of inner product. Spell out
what this means over R (definite, symmetric, bilinear) and C
(definite, conjugate symmetric, sequilinear). Norm,
distance and orthogonality. Examples: dot product; L^2
inner product on C^0[a,b].
[notes: bottom of page 19]
Lecture 10: Example: square summable sequences.
Cauchy-Schwarz inequality, examples thereof. Pythagoras and
parallelogram identities; triangle inequality.
[notes: end of section 3.1]
Lecture 11: Orthonormal lists of vectors and orthonormal
bases. Easy to compute coefficients in expansion with
respect to orthonormal basis. Parseval and Bessel.
Gram-Schmidt orthogonalisation. Cor: orthonormal bases
exist.
[notes: end of Cor 3.8]
Lecture 12: Explicit example of Gram-Schmidt. Defn of
orthogonal matrix. QR decomposition of an invertible matrix
and how to compute same.
Defn of orthogonal complement of U<=V. Elementary
properties.
[notes: top of page 26]
Lecture 13: Orthogonal complement is an actual complement when U
finite-dimensional. Dimension of orthogonal complement.
Orthogonal projection and properties. Application:
orthoprojection onto a subspace gives nearest point on
subspace. Chat about approximating sin by degree 5
polynomials.
[notes: End of chapter 3]
Lecture 14: New Chapter: linear operators on inner product
spaces. Defn of linear operator. Reminder about linear
maps and matrices in presence of a basis. Defn of adjoint
of an operator. Examples of operators that have adjoints.
Thm: linear operators on a finite-dimensional inner product
space have a unique adjoint.
[notes: line 8 of page 31]
Lecture 15: Formula for matrix of adjoint
with respect to orthonormal basis. Self/skew-adjoint
operators; (skew-)Hermitian/symmetric matrices. Examples.
Linear isometries: definition. Linear isometries of a
finite-dimensional inner product space characterised by
adjoint=inverse.
[notes: remark following (4.4), page 32]
Lecture 16: Orthogonal/unitary transformations/matrices:
defn. General linear group is a group and the
orthogonal/unitary group is a subgroup. Classification of
rigid motions of a real vector space (sketch proof).
[notes: End of (4.6), page 35]
Lecture 17: Revision of eigenvalues and eigenvectors.
Invariant subspaces. Commuting operators preserve each
others eigenspaces. If U is f-invariant, U^\perp is
f^*-invariant. Defn of normal operators; examples. Normal
operators preserve the perp of their eigenspaces.
[notes: Bottom of page 36]
Lecture 18: Diagonalisable and orthogonally diagonalisable
operators. When is an operator orthogonally diagonalisable?
Necessary condition. Spectral thm for normal operators.
[notes: end of section 4.2.3, page 38]
Lecture 19: Properties of self-adjoint operators. Real
self-adjoint operators have eigenvalues. Spectral thm for
real self-adjoint operators. Chat about inner product
spaces and self-adjoint operators in quantum mechanics.
Spectral theorem for symmetric/Hermitian matrices.
[notes: end of (4.17), page 40]
Lecture 20: Example of matrix spectral theorem. Formula for
self-adjoint f in terms of eigenbasis and eigenvalues.
Properties of f^*f for arbitrary operator f: non-negative
eigenvalues and same kernel as f. Singular values and the
singular value decomposition.
[notes: End of Chapter 4]
Lecture 21: New chapter: Duality. Defn of dual space.
Examples of linear functionals. Dual basis of V^* to one of
V. Dual basis as coordinate functions. dim V=dim
V^*. Riesz Repn Thm. Example of Riesz in action.
[notes: End of example at top of page 45.]
Lecture 22: Sufficiency Principle. All bases of V^* are
dual bases. Evaluation is a linear injection V to V^** and
so an isomorphism in the finite-dimensional case. Dual
space as set of linear equations. Solution sets: defn.
[notes: end of page 46.]
Lecture 23: Solution sets: dimension. Criterion for linear
functionals to span V^* and application. Properties of
solutions sets under inclusion, sum and
intersection. Annihilators: defn, dimension and properties
under inclusion, sum and intersection.
[notes: end of (5.11) near top of page 49]
Lecture 24: Relation between sol and ann: mutually inverse
when V finite-dimensional. Transposes: a linear map W^*\to
V^* from a linear map V \to W. Examples. Rant about
category theory. Transposes and matrices.
[notes: end of (5.15), bottom of p. 50]
Lecture 25: Relation between kernels and images of a map and
its transpose. Application: f injects/surjects if and only
if f^T surjects/injects (for finite-dimensional domain and
target). New chapter: bilinearity. Defn of bilinear map,
pairing and form. Examples.
[notes: end of example 4 on page 52.]
Lecture 26: Matrices give bilinear maps on F^mxF^n.
Bilinear forms and matrices, change of basis formula,
congruent matrices. Symmetric bilinear forms: defn and
equivalence to symmetry of matrices. Radical and rank of a
symmetric bilinear form. Interpretation via induced map V
to V^*.
[notes: middle of page 54.]
Lecture 27: Matrix of linear map. Examples. Quadratic forms
and their polarisation. How to compute same. Example.
[notes: end of section 6.2.3, page 56.]
Lecture 28: Diagonalisation theorem. Alternative approach
via spectral thm. What it means for quadratic forms.
How to find a diagonalising basis.
[notes: top of page 58.]
Lecture 29: Example of diagonalising : three methods. Signature
of a quadratic form. Inertia Theorem to compute same (statement).
Example of use.
[notes: pages 59-60.]
Lecture 30: Inertia Theorem (proof). Summary of situation
over R and C.
[notes: end of chapter 6]
THAT'S ALL FOLKS!