vector space

Vectors are mathematical objects that describe quantities with both magnitude and direction, defined in Euclidean spaces and studied through their algebraic properties, operations, and geometric interpretation.

A vector \( b \in \mathbb{R}^n \) is said to be a linear combination of the vectors \( v_1, v_2, \dots, v_k \in \mathbb{R}^n \) if and only if there exist scalars \( c_1, c_2, \dots, c_k \in \mathbb{R} \) such that:

\[
b = c_1 v_1 + c_2 v_2 + \dots + c_k v_k
\]

Kernel and image of a linear map are the two fundamental subspaces associated with a linear transformation. The kernel contains all vectors mapped to the zero vector, while the image consists of all vectors that can be obtained as outputs of the transformation.