Linear Algebra I

Office hours: by appointment. S125 (Malostranské nám. 25, 1st floor).

Consultation hour: Wednesday 15 November 13:00, Troja building T, seating upstairs from main entrance.

Examination

Examinations will take place in Malostranske namesti 25 (beginning S125, 1st floor).

Checklist for revision

Lectures

Time: Wednesdays 09:00 - 10:20 (on Dean's Day, 8 November, there will be no lecture)

Place: T5 (T149, Troja OP - 1.patro/1st floor. V Holešovičkách 747/2, 180 00 Praha 8)

Course syllabus (indicative of what is to come, not necessarily to the minute detail of what actually will be covered - this in the fullness of time...)

Resources

David Poole, Linear Algebra, A Modern Introduction, 3rd Int. Ed., Brooks Cole, 2011.

Essence of Linear Algebra (3Blue1Brown YouTube series)

C. D. Meyer. Matrix analysis and applied linear algebra. SIAM, Philadelphia, PA, 2000.

Robert Beezer, A First Course in Linear Algebra - a free online textbook.

MIT's website includes video lectures and supplementary material to Gilbert Strang's Introduction to Linear Algebra. Strang's own sample material etc.

Pavel Klavík's orgpad map of Linear Algebra

Jiri Matousek's preliminary version of the recommended book Thirty-three miniatures: mathematical and algorithmic applications of linear algebra.

M. Anderson and T. Feil, A First Course in Abstract Algebra: Rings, Groups, and Fields, CRC, 2015 (chapters 17-19 for groups).

F. Goodman, Algebra: Abstract and Concrete, edition 2.6

Permutations (Cut-the-Knot.org)

Exercises

Classes, on Wednesdays at 15:40 in T5, are led by Ondřej Pangrác.

The online collection of exercises provides practice problems, many with solutions.

Material covered

4 October

Points as locations, vectors as displacements. Vectors in the plane. Position (tail), direction and length. Standard position (tail at origin). Column vector notation. Vector addition (head-to-tail rule, parallelogram diagonal). Scalar multiplication. Negative. Vector subtraction (other parallelogram diagonal). Vector space R^n. Components. Componentwise addition and scalar multiplication. Vector space axioms [Poole, Theorem 1.1] informally introduced as properties of R^n over R. Proof by appeal to axioms for R and using formal definitions of addition and scalar multiplication.

11 October

Parallelogram rule for vector addition and subtraction, parallelepiped with three edge directions/lengths u, v, w to see why vector addition is associative, i.e. (u+v)+w = u+(v+w). Review of vector space as closed under vector addition and scalar multiplication, together with axioms. Linear combinations of vectors in R^2. Two vectors span the whole plane when not parallel, i.e. one not a scalar multiple of the other: detailed proof. Parametric equation of a straight line (point on line and scalar multiple of direction vector). Intercept-intercept form of equation for a straight line. System of linear equations in two variables.

18 October

Span of a set of vectors. Paremetric equation for lines and planes. Linear independence. Length (norm) and angle, dot product. Properties of dot product [Poole, Theorem 1.2] and length [Poole, Theorem 1.3]. Normalizing a vector to a unit vector. Standard unit vectors (coordinates = Cartesian coordinates). Cauchy-Schwarz Inequality (proof for R^2 only). Triangle Inequality (statement and intuition only).

25 October

Brief look at: Law of cosines; dot product of normalized vectors equals angle between them. Pythagoras' Theorem. Orthogonal vectors. Normals to a line, plane. Projection of a vector onto another vector. Equations of line and plane by normal vector. [Poole, Ch. 1.2, 1.3]

Start: Systems of linear equations [Poole, Ch. 2.1]

1 November

Systems of linear equations in n variables, solution to a linear equation, solution of a system of linear equations, augmented matrix, elementary row operations, row reduction to echelon form, back substitution, row equivalence and preservation of solution set, Gaussian elimination, leading variables and free variables. [Poole, Ch. 2.2]

8 November - NO LECTURE (Dean's Day)

Read Poole Section 2.2, reviewing last lecture's material plus: rank of a matrix, Gauss--Jordan elimination to reduced row echelon form, homogenous systems, intersections of hyperplanes, linear dependence. Read Poole Section 2.3 as preparation for next lecture.

15 November

Reduced row echelon form, matrix multiplication applied to a vector (dot products by rows, linear combination of columns), elementary row elimination operations as matrix multiplications, general matrix multiplication (three forms: entries as dot products, rows as linear combinations, columns as linear combinations) [Strang, Chapter 1, sections 1.4, 1.5]

22 November

Matrix multiplication (associative, not usually commutative, distributive), permutation matrices.

[Strang, Chapter 1, sections 1.4, 1.5; Poole, Chapter 3, sections 3.1-3.4.]

29 November

Transpose and inverse of a matrix, elementary matrices, upper and lower triangular form, LU-factorization of a matrix, reduced row echelon form and Gauss-Jordan inversion of a matrix.

6 December

Gauss-Jordan algorithm using mix of numerical and variable entries in 2x2 and 3x3 matrices. Groups and fields. Integers modulo m a field iff m prime. Matrices over other fields than reals. Inverting numerical examples modulo m by Gauss-Jordan algorithm.

13 December

Little Fermat theorem for integers modulo a prime. Matrices over a field. Transpose and inverse properties. Permutation matrices (inverse equals transpose).

Vector spaces and subspaces. Linear combinations and span. Row, column and null spaces of a matrix.

[Poole, Sections 3.5, 6.1, 6.2]

20 December

Bases, bases same size (dimension). Rank of matrix, rank-nullity theorem.

3 January

Steinitz Exchange Lemma. Testing whether a set of vectors forms a basis. Coordinates relative to a basis. Various equivalent statements to "A is an invertible matrix".

10 January

Linear transformations, Change of basis. [Poole, Sections 3.6, 6.3, 6.4]