Linear Alegbra and Vector Spaces

Vector space (over a field $F$)

A set $V$ equipped with vector addition and scalar multiplication satisfying eight axioms, including associativity, commutativity of addition, existence of identity and inverses, and distributivity.

Subspace

A subset $W$ of a vector space $V$ that contains the zero vector and is closed under vector addition and scalar multiplication.

Linear combination

An expression formed by multiplying each vector in a set by a scalar and adding the products together: $c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_k \mathbf{v}_k$.

Span

The set of all possible linear combinations of a subset of vectors $S$, denoted $\text{span}(S)$, forming the smallest subspace containing $S$.

Linear independence

A property of a set of vectors where the equation $c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_k \mathbf{v}_k = \mathbf{0}$ holds only when all scalars $c_i = 0$.

Basis

A linearly independent subset of a vector space that spans the entire vector space.

Dimension

The unique cardinality (number of vectors) in any basis of a finite-dimensional vector space.

Linear transformation

A mapping $T: V \rightarrow W$ between vector spaces satisfying $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{u}) = cT(\mathbf{u})$ for all vectors and scalars.

Kernel (Null space)

The set of all vectors $\mathbf{v} \text{ in } V$ mapped to the zero vector by a linear transformation: $\{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}\}$.

Rank-Nullity Theorem

The theorem asserting that for any linear map $T: V \rightarrow W$ with finite-dimensional domain $V$, $\dim(\text{null}(T)) + \dim(\text{range}(T)) = \dim(V)$.