Vector Spaces Flashcards
What is a field?
A set F of numbers with the property that if a and b are in F, so are a+b, a-b, a*b and a/b (b is not 0)
Basically a set that has 2 operations, one with identity and inverse, and the other with identity and inverse for all but a single element
What does a vector space consist of?
A set V, made up of vectors (usually in R^2 or R^3)
A field F, made up of scalars (usually in R)
V is a commutative group under addition (doesn’t matter which way round you write the vectors)
Vector addition operation which takes two vectors and outputs a third vector
Scalar multiplication operation which takes a scalar and a vector, and multiplies them to produce a vector
What are the vector space axioms?
Associativity of multiplication: a(bv) = ab(v)
Unitarity: 1v = v
Distributivity: a(u+v) = au + av, (a + b)v = av + bv
What is the linear combination of vectors v1 + v2 + …. + vn and scalars a1 + a2 + ….. + an
a1v1 + a2v2 + …. + anvn
What is the set R^n
The set consisting of all n-tuples of real numbers
(0, 1, 3) and (4, 2, 0) are in R^3 for example
What is the span of a set of vectors?
The set of all its possible linear combinations, often infinite if the scalar field is infinite (e.g. integers)
The span of the set {(0, 1, 0), (0, 1, 2)} contains 2(0, 1, 0) + 5(0, 1, 2), 3(0, 1, 0) + 10(0, 1, 2) etc.
All linear combinations in that span are vectors in the form (0, a+b, 2b), forming a plane
What is the spanning set for a vector space (F, V)?
A set of vectors whose span is V
if the span of S is V, then S spans V
What does it mean when a set of vectors is linearly dependent?
There exists a linear combination of these vectors which equals the zero vector (the zero vector is in the span)
At least one of the scalars must be non-zero
For example, 5a + 6b - 3c = 0
This means you can also say that c = 5/3a + 2b, you can write c in terms of the others
What is the basis of a vector space?
A subset S of a vector space V is a basis for V if the span of S is V
All the vectors in S to be linearly independent
