Vector spaces

Vector spaces

A vector space consists of the following:

1. a field G of scalars;
2. a set V of objects, called vectors
3. a rule called vector addition, which associates with each pair of vectors u, v in V a vector u + v in V, called the sum of u and v such that
   (a) addition is commutative, u + v = v + u
   (b) addition is associative, u + (v + q) = (u + v) + q
   (c) there is a 0 vector in V, such that u + 0 = u for all u in V.
   (d) for each vector u in V, there is a unique vector - u in V such that u + (- u) = 0
4. a rule called scalar multiplication, which associates each scalar k in F and vector u in V a vector ku in V, called the product of k and u, in such a way that
   (a) 1 * u = u for every u in V;
   (b) (k₁k₂) u = k₁(k₂ u);
   (c) k (u + v) = ka + kb;
   (d) (k₁ + k₂)u = k₁u + k₂u.

Subspaces

Bases and dimension

Coordinates

Summary of row-equivalence

Computations concerning subspaces