Summary

Vector space

Definition

Definition

A vector space is ordered pair $(V,F)$ where $V$ is a set and $F$ is a field, together with two operations:

• Vector addition: $+:V\times V\to V$
• Scalar multiplication: $\cdot :F\times V\to V$

Such that the following properties hold:

• $(V,+)$ is an Abelian group. Here, $0$ is the additive identity.
• Scalar multiplication is distributive over vector addition: $a\cdot (v+w)=a\cdot v+a\cdot w$ for all $a\in F$ and $v,w\in V$.
• Scalar multiplication is distributive over field addition: $(a+b)\cdot v=a\cdot v+b\cdot v$ for all $a,b\in F$ and $v\in V$.
• Scalar multiplication is associative: $(a\cdot b)\cdot v=a\cdot (b\cdot v)$ for all $a,b\in F$ and $v\in V$.
• Scalar multiplication by the multiplicative identity of the field is the identity operation on the vector space: $1\cdot v=v$ for all $v\in V$.

Examples

Example

• The set of real numbers ${\mathbb{R}}^n$ with standard addition and scalar multiplication is a vector space.
• The set of $n\times m$ matrices over a field is a vector space.
• The set of polynomials of degree at most $n$ is a vector space.

Subspaces

Definition

A subspace of a vector space $V$ is a subset $U$ of $V$ that is itself a vector space with the operations of vector addition and scalar multiplication inherited from $V$. The subspace is denoted by $U\subset V$.

Examples

• The set of all polynomials of degree at most $n$ is a subspace of the vector space of all polynomials.
• ${\mathbb{R}}^n$ is a subspace of ${\mathbb{C}}^n$

Span

Definition

The span of a set of vectors $\{v_1,v_2,\dots ,v_n\}$ is the set of all linear combinations of the vectors. It is denoted by $\text{span}(v_1,v_2,\dots ,v_n)$.

Linear combinations

Definition

A linear combination of a set of vectors $\{v_1,v_2,\dots ,v_n\}$ is ${\sum }_{i=1}^n{\lambda }_i{v}_i$ where ${\lambda }_i$ are scalars.

Basis

Definition

A basis of a vector space $V$ is a set of linearly independent vectors that span $V$.

Important

• A basis is a minimal set of vectors that can represent all vectors in the vector space.
• A basis is a maximal set of linearly independent vectors.

Linear independence

Definition

A set of vectors $\{v_1,v_2,\dots ,v_n\}$ is linearly independent if the only solution to the equation ${\sum }_{i=1}^n{\lambda }_i{v}_i=0$ is ${\lambda }_i=0$ for all $i$.

Dimension

Definition

The dimension of a vector space $V$ is the number of vectors in a basis of $V$. It is denoted by $\dim (V)$.

Linear transformations

Definition

A linear transformation $\mathcal{T}:V\to W$ between two vector spaces $V$ and $W$ is a function that preserves vector addition and scalar multiplication. It satisfies the following properties:

• $\mathcal{T}(u+v)=\mathcal{T}(u)+\mathcal{T}(v)$ for all $u,v\in V$.
• $\mathcal{T}(a\cdot v)=a\cdot \mathcal{T}(v)$ for all $a\in F$ and $v\in V$.

Kernel

Definition

The kernel of a linear transformation $\mathcal{T}:V\to W$ is the set of all vectors in $V$ that are mapped to the zero vector in $W$. It is denoted by $\ker (\mathcal{T})$.

Image

Definition

The image of a linear transformation $\mathcal{T}:V\to W$ is the set of all vectors in $W$ that are the result of applying $\mathcal{T}$ to some vector in $V$. It is denoted by $\text{im}(\mathcal{T})$.