1 vector spaces

V - vector space over some field \mathbb{K} if:

• for every u, v, w \in V we have u + v = v + u and u + (v + w) = (u + v) + w
• there is \mathbf{0} \in V called the zero vector such that for any v \in V we have v + 0 = 0 + v = v
• for ever v \in V there is -v \in V such that v + -v = -v + v = 0

1. (\forall v \in V) 0 \cdot v = \mathbf{0}
2. (\forall a \in F) a \cdot \mathbf{0} = \mathbf{0}
3. (\forall a \in F)(\forall v \in V) (-a)v = a(-v) = -(av)
4. (\forall a \in F)(\forall v \in V) av = \mathbf{0} \implies a = 0 \lor v \in \mathbf{0}

1.0.1 subspaces

A subset W \subseteq V is called a subspace of V if W is a vector space over \mathbf{K} under the same operations of vector addition and scalar multiplication.

1.0.2 linear combination

Let a_1, \cdots, a_n \in \mathbf{K} and v_1, \cdots, v_n \in V. The vector a_1v_1 + \cdots + a_nv_n \in V is called a linear combination.

1.0.3 span

A span of S \subseteq V is a set of all linear combinations of S. Let S \subseteq V then span(S) \subseteq V or span(S) = \{a_1v_1 + \cdots + a_nv_n : a_i \in \mathbb{K} and v_i \in S\}.

1.0.4 linear independence

a_1v_1 + \cdots + a_nv_n = \mathbf{0} \implies a_1 = \cdots = a_n = 0
otherwise it is called linearly dependent

1.0.5 basis

We call B \subseteq V a basis if:

• B is linearly independent
• span(B) = V

1.0.6 dimension

Maximal amount of vectors that are together linearly independent. Size of a basis: \dim(V) = |B|. For example: \dim(\mathbb{R}^3) = |\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}| = 3

Let V be a vector space and \dim(V) = n. Finally, let S \subseteq V. Then:

• if span(S) = V then |S| \ge n
• if S is linearly independent and |S| = n then S is a basis
• if span(S) = V and |S| = n then S is a basis
• if |S| > n then S is linearly dependent