MATH 223 Midterm 1 Definitions

Field

A set F with two operations: addition denoted by + and multiplication denoted by *. The set is closed under those operations, meaning a+b and a•b are in F for any a,b in F. The following axioms must be satisfied:

Associativity: (a+b)+c = a+(b+c), (a*b)*c=a*(b*c)

Commutativity: a + b = b + a, a*b = b*a

Neutral Elements: For addition the neutral element is called zero, for multiplication it is called one.

Inverse Elements: Additive inverse is -a, multiplicative inverse is a^(-a)

Distributivity: Multiplication is distributive over addition. a*(b+c)=(a*b)+(a*c)

Non-triviality: The additive and multiplicative identities are distinct 0 ≠ 1

Vector Space

A vector space of F is a set V equipped with two operations: vector addition, scalar multiplication. And satisfy all the axioms of a field.

Zero Vector

Additive identity: v + 0 = v

Subspace

A subspace of a vector space V is a subset W ⊆ V that is itself a vector space under the same operations of addition and scalar multiplication. Must include zero vector and be closed under addition and multiplication.

Linear Combination

A linear combination of vectors v₁, v₂, …, vₙ in a vector space V is any vector that can be written in the form c₁v₁+c₂v₂ + … + cₙvₙ. Where c₁, c₂, …, cₙ are scalars.

Span of a set (and also what it means that a set S spans the entire space V)

The span of S is the set of all linear combinations of those vectors: span(S) = {c₁v₁ + c₂v₂ + … + cₙvₙ | c₁, c₂, …, cₙ ∈ F}.

Every vector in V can be written as a linear combination of the vectors in S. So the vectors in S are “enough” to reach anywhere in the space.

Linearly dependent set

A set of vectors in a vector space V is linearly dependent if there exist scalars c₁, c₂, …, cₙ, not all zero, such that c₁v₁+c₂v₂+…+cₙvₙ=0

Linearly independent set

Not linearly dependent (only equals zero when all coefficients are zero)

Basis

A set of vectors B = {v₁, v₂, …, vₙ} in a vectors space V is a basis of V if it satisfies two conditions:

1. Spanning: The set B spans V, i.e., every vector in V can be written as a linear combination of the vectors in B.

2. Linear independence: the vectors in B are linearly independent, i.e., the only solution to c₁v₁+c₂v₂+…+cₙvₙ=0 is c₁=c₂=cₙ

Every basis of a vector space has the same number of vectors.

Dimension

The dimension of a vector space V, denoted dim(V), is the number of vectors in any basis of v.

Every subspace is a

Vector space

If a set S is linearly dependent, then there exists a vector v ∈ S that can be written as a

Linear combination of the other vectors from S

The cardinality of any linearly independent set in a vector space V is at most

the cardinality of any spanning set of V

Any linearly independent set in V can be

completed to a basis

Any two bases

have the same cardinality

A set S is a basis of V if and only if

every vector in V can be written as a linear combination of the vectors in S in a unique way.