fundamental theorem of calculus part 1 practice problems

Scalar dilations and rotations combine to form a copy of the field of complex numbers. Rotations and reflections of the plane combine to form the two-dimensional orthogonal group. Linear transformations are introduced, focusing on transformation of the plane. Diagonalisation is a manifestation of the general Conjugation Principle, explored in different contexts. In practice, there are iterative numerical techniques for finding approximations of eigenvalues and eigenvectors, using a technique associated with Perron’s Theorem. Associated eigenvectors can then be found by solving an associated homogeneous system of equations. Eigenvalues can be found, in principle, by solving the characteristic equation of a matrix. This has many applications, including, for example, in exploring the behaviour of a Markov process described by a stochastic matrix, the mathematics of which underlies search engines on the internet. This leads to the technique of diagonalisation, enabling one to solve difficult problems in matrix arithmetic, including finding formulae for powers of a given square matrix. Interpreted geometrically, these allow one to find directions in which a given linear operator associated with a matrix moves vectors in straight lines. This module continues the development of matrix arithmetic by introducing eigenvalues and associated eigenvectors.

see a brief introduction to the arithmetic of complex numbers and discussion of the Fundamental Theorem of Algebra (fourth week).

see applications to simple Markov processes and stochastic matrices, and an introduction to linear transformations, illustrated using dilation, rotation and reflection matrices (fourth week),.

be introduced to the theory of eigenvalues and eigenvectors, how they are found or approximated, and their role in diagonalisation of matrices (fourth week),.

be introduced to and gain familiarity with matrix arithmetic, matrix inverses, the role of elementary matrices and their relationships with matrix inversion and systems of equations, calculations and theory involving determinants (third week),.

develop fluency with lines and planes in space, represented by vector and Cartesian equations, and learn how to solve systems of equations, using the method of Gaussian elimination and introduction of parameters, using fields of real numbers and modular arithmetic with respect to a prime number (second week),.

gain familiarity with the arithmetic of geometric vectors, which may be thought of as directed line segments that can move about freely in space, and can be combined in different ways, using vector addition, scalar multiplication and two types of multiplication, the dot and cross product, related to projections and orthogonality (first week),.Students taking Introduction to Linear Algebra will: This course emphasises geometric intuition, gradually introducing abstraction and algebraic and symbolic manipulation, while at the same time striking a balance between theory and application, leading to a mastery of key threshold concepts in foundational mathematics. The key ideas involve vector and matrix arithmetic as well as clever methods for working around or overcoming difficulties, a form of obstacle avoidance, articulated in this course as the Conjugation Principle. It is recommended that participants in the MOOC Introduction to Linear Algebra have already taken, or take in parallel, the MOOC Introduction to Calculus.Īll of our modern technical and electronic systems, such as the internet and search engines, on which we rely and tend to take for granted in our daily lives, work because of methods and techniques adapted from classical linear algebra.

Over time, students discover that linear algebra and calculus are inseparable (but not identical) twins that interlock to form the backbone of almost all applications of mathematics to physical and biological sciences, engineering and computer science. They are studied by almost all mathematics students at university, though typically labelled as different subjects and taught in parallel. Linear algebra and calculus are the two most important foundational pillars on which modern mathematics is built.

Fundamental theorem of calculus part 1 practice problems