## 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.
• Examples.

(Lec 7) Tue 16 Feb

• Subspaces and linear maps.
• Examples.
• Linear isomorphism.
• Example: invertible matrices.
• Linear bijections are isomorphisms.

(Lec 8) Wed 17 Feb

• Kernels and images; they are subspaces
• Examples.
• 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

• Quiz: Ex 4.8

(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

(Lec 20) Tue 5 Apr

• A invertible iff det A non-zero
• Cramer's rule

(PC 4) Wed 6 Apr

• Quiz: Ex 8.7

(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

• Quiz: Ex 10.7.