Handout 01 - Introduction to Vector Spaces (4).pdf -...

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MATH1231 Mathematics 1B – Algebra Lecture 1: Introduction to Vector Spaces Lecturer: Sean Gardiner – [email protected]
Vectors (MATH1131 revision) In MATH1131, we saw a vector is a mathematical object that can be thought of geometrically as something with direction and magnitude, or algebraically as an element of R n (an ordered vertical list of n real numbers). We can add any two vectors to produce another vector, written v + w , and this vector addition is associative and commutative. We can also multiply any vector by a scalar (real number) to produce another vector, written λ * v or just λ v , and this scalar multiplication is associative. Furthermore, vector addition and scalar multiplication together are distributive. There is also the concept of a zero vector 0 which has the property that any vector added to it goes unchanged. The above properties are not unique to elements of R n . Another set of mathematical objects that satisfy all the above properties is the set of matrices of a particular size, for example all 2 × 3 matrices. This makes sense, since we can consider any element of R n as an n × 1 matrix. Similarly, another set of mathematical objects that satisfies all the above properties is the set of complex numbers C . This makes sense, since we know numbers plotted in the complex plane correspond with position vectors in R 2 . Mathematics 1B – Algebra Lecture 1: Introduction to Vector Spaces 1/20
Vector spaces We would like to be able to describe all these systems under a single unifying concept. The term given to any such sort of system is a vector space , and its elements are called vectors. In this more abstract sense of a vector space, the term “vector” no longer needs to be so restrictive and will therefore not always refer to elements of R n . For a vector space to be well-defined, we must provide four pieces of information: The set of vectors V . (This may be R n or something else entirely.) The set of scalars F . (This is typically either R or C .) The vector addition operation + . The scalar multiplication operation * . (This symbol is often omitted in practice, so λ * v is usually written as λ v .) To properly reference a vector space, we should provide the list ( V , +, * , F ) . In practice, once the operations and set of scalars has been established, we refer to the vector space just as V . Mathematics 1B – Algebra Lecture 1: Introduction to Vector Spaces 2/20
Field axioms (MATH1131 revision) Before we proceed with investigating the properties of a vector space, we should establish the properties of a set of scalars, which is also known as a field . Recall from MATH1131 that Q , R and C are fields, and they can each be described as a set F satisfying the following axioms: Closure under addition: x + y F for all x , y F . Additive associativity: ( x + y ) + z = x + ( y + z ) for all x , y , z F .
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