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.
Examinations will take place in Malostranske namesti 25 (beginning S125, 1st floor).
Checklist for revision
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)
(indicative of what is to come, not necessarily to the minute detail of
what actually will be covered - this in the fullness of time...)
David Poole, Linear Algebra, A Modern Introduction, 3rd Int. Ed., Brooks Cole, 2011.
Essence of Linear Algebra
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.
includes video lectures and supplementary material to Gilbert
Strang's Introduction to Linear Algebra. Strang's own sample material etc.
'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
Classes, on Wednesdays at 15:40 in T5, are led by Ondřej Pangrác
The online collection of exercises
provides practice problems, many with solutions.
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
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
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
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).
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]
Systems of linear equations [Poole, Ch. 2.1]
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.
- NO LECTURE (Dean's Day)
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.
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]
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.]
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.
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.
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]
Bases, bases same size (dimension).
Rank of matrix, rank-nullity theorem.
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".
Linear transformations, Change of basis. [Poole, Sections 3.6, 6.3, 6.4]