Ma10210 Algebra 1B
Diary of Lectures 2014/15 (Semester 2)
INTRODUCTION TO LINEAR ALGERBRA
(Lec 1) Mon 1 Feb
- Ch 1. Linear systems and matrices
- Examples of linear systems.
- Three views on interpretation.
(Lec 2) Tue 2 Feb
- Gaussian elimination.
- Row echelon form (REF and RREF), elementary row operations (EROs).
(Lec 3) Wed 3 Feb
- Recap on fields, vectors and matrices.
(Lec 4) Mon 8 Feb
- matrices are linear maps F^n -> F^m
- linear maps F^n -> F^m are matrices
- invertible matrices
- their elementary properties
(Lec 5) Tue 9 Feb
- why Gaussian elimination works:
EROs are multiplication by invertible matrices
- computation of matrix inverses by Gauss-Jordan method,
i.e. reduction to RREF
- the null space of a matrix; it's a linear subspace
- basis of a linear subspace (temp.def.)
- Gaussian elimination provides a basis of the null space
(PC 1) Wed 10 Feb
- rank of a matrix (temp.def.)
- Quiz: Ex 2.8
(Lec 6) Mon 15 Feb
- Ch 2. Vector spaces and linear maps
- Definition of abstract vector space and basic properties.
(Lec 7) Tue 16 Feb
- Subspaces and linear maps.
- Linear isomorphism.
- Example: invertible matrices.
- Linear bijections are isomorphisms.
(Lec 8) Wed 17 Feb
- Kernels and images; they are subspaces
- Ker=0 iff injective; fibres are cosets of Ker.
(Lec 9) Mon 22 Feb
- linear span of vectors
- spanning and independent sets, bases.
- bases and coordinates, i.e. isomorphism with F^n.
- matrix representing a linear map
(Lec 10) Tue 23 Feb
- (square) matrix representing a linear operator
- equivalent and similar matrices, and their interpretation
- change of basis matrix
- matrices representing the same map/operator are equivalent/similar
(PC 2) Wed 24 Feb
(Lec 11) Mon 29 Feb
- Ch 3. Dimension and Rank
- Defs: finite dimensional vector space, dimension
- Fundamental Lemma: a matrix with trivial null space
has #cols <= #rows
- Cor: invertible matrices are square
- Thm: # independent list <= # spanning list;
hence all bases have the same size.
(Lec 12) Tue 1 Mar
- Prop: shortening spanning lists and extending independent lists
- Prop: minimal spanning lists and maximal independent lists are bases
(Lec 13) Wed 2 Mar
- Thm: subspaces of fin.dim.v.s. are fin.dim.
- Thm: every finite spanning list contains to a basis and
(in a fin.dim.vec.spc.) every independent list extends to a basis
- Example: finding bases of column spaces, by sifting.
(Lec 14) Mon 7 Mar
- Def: rank (of linear map/matrix)
- rank is invariant under equivalence
- row rank = column rank
(Lec 15) Tue 8 Mar
- Def: nullity
- Rank-Nullity Theorem
- properties of maximal rank linear maps
- direct sums
(PC 3) Wed 9 Mar
- dim of sum and intersection
- Quiz: Ex 6.7
(Lec 16) Mon 14 Mar
- Ch 4. Determinants and Inverses
- Sum formula for det
- det for triangular and block triangular matrices
- det A transpose = det A
(Lec 17) Tue 15 Mar
- det is multilinear and alternating as function of columns (or rows)
- change of det under ECOs (or EROs)
(Lec 18) Wed 16 Mar
- characterisation theorem :
any multilinear alternating function is a multiple of det
- product formula : det AB = det A det B
- determinant of a linear operator
(Lec 19) Mon 4 Apr
- Minors and row/column expansion of det
- Adjugate and (adj A) A = (det A) I
(Lec 20) Tue 5 Apr
- A invertible iff det A non-zero
- adjugate formula for inverse
- Cramer's rule
(PC 4) Wed 6 Apr
(Lec 21) Mon 11 Apr
- Ch 5. Eigenvalues and diagonalisation
- Eigenvalues, eigenvectors, eigenspaces.
- Characteristic polynomial.
- Prop: the eigenvalues are the roots of the char.poly.
(Lec 22) Tue 12 Apr
- Diagonalisability of operators/matrices.
- Existence of eigenbasis vs. similarity to diagonal matrix.
- Calculation of powers, e.g. for solving linear recurrences.
- Example: Fibonacci sequence.
(Lec 23) Wed 13 Apr
- Examples of non-diagonalisable matrices.
- Thm: independence of eigenvectors with distinct eigenvalues.
(Lec 24) Mon 18 Apr
- Algebraic and geometric multiplicity of e'values.
- alg.mult >= geom.mult.
- Thm: A diagonalisable iff char.poly.(A) has linear factors
and a.m.=g.m. for all e'values
(Lec 25) Tue 19 Apr
- Application: understanding quadratic forms on R^2
using e'values and e'vectors
(PC 5) Wed 20 Apr
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