Linear Algebra Done Right
**Sheldon Axler – **
Chapter 1: Vector Spaces
Complex Numbers
Complex numbers were invented so we could take square roots of negative numbers.
Complex numbers are defined as an ordered pair of real numbers a and b, and are typically written as . One can even think of the set of real numbers as being a subset of the set of complex numbers, where for all real numbers. Note that a complex number is considered a single number, even though it involves an ordered pair of real numbers.
All the typical arithmetic operations and properties apply to complex numbers (commutativity, associativity, etc), so you can work with them fairly similarly.
Multiply complex number as so: The calculation for multiplying complex numbers doesn’t need to be memorized, as it can always be re-derived as needed. That said, if you need some intuition, notice that , so when you multiply by , that becomes .
Fields
A field is a set of at least two distinct elements along with various operations of addition and multiplication. Examples of fields include the set of all real numbers and the set of all complex numbers.
Vector Spaces
A vector space is a set of numbers with addition and multiplication that have the typical properties we associate with those operations. Elements of a vector space are called vectors.
We can define addition or scalar multiplication as the set as functions that respectively maps pairs of numbers within the set to a new numbers also within the set.