vectors
(1.2 hours to learn)
Summary
A vector is is a fundamental mathematical structure that is characterized by both a direction (ordering) and a magnitude. For instance, wind has both a direction (North, South-West, etc) and a magnitude (10 km/hour) and could be represented as a vector (10 km/hour South-West). A point in Euclidean space is often represented as a vector of its coordinates and is the most common type of vector encountered. More generally, a vector is an element of a vector space -- a set that is closed under scalar multiplication and vector addition. [additional note: a vector is a very general entity that takes on many forms depending on its context, for instance, in certain vector spaces a vector could be a function such as f(x) = sin(x)]
Context
-this concept has no prerequisites-
Core resources (read/watch one of the following)
-Free-
→ Khan Academy: Linear Algebra
-Paid-
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location:
Section 1.1, "Vectors in R^n", pages 1-6
→ Introduction to Linear Algebra
An introductory linear algebra textbook with an emphasis on numerical methods.
Location:
Section 1.1, pages 2-6
Supplemental resources (the following are optional, but you may find them useful)
-Free-
→ A First Course in Linear Algebra (2012)
A linear algebra textbook with proofs.
Additional dependencies:
- linear systems as matrices
→ Wolfram MathWorld
→ MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
See also
- Central linear algebra concepts which build directly on vectors include: Using a small set of axioms, the basic properties of vectors can be generalized to other vector spaces ; examples include functions and polynomials.