Overview

A vector space $V$ is a set of objects, called vectors, that follow a set of rules under the operations of addition ($+$) and scalar multiplication ($\cdot$).

Note that these operations do not have to technically be the same under the real numbers (for example), rather we go for more abstract operations.

• An addition on a set $S$ is a function that assigns an element $u+v \in S$ to two elements $u,v \in S$. Similarly
• A scalar multiplication on a set $S$ is a function that assigns an element $\lambda v \in S$ for each element $\lambda \in \R$ and each $v \in S$ (note that in certain cases we can say $\lambda \in \mathbb{C}$ or another set, depending on where we want to work; we will assume from here on that we use $\R$ unless specified otherwise)

The Criteria

• Associativity and commutativity hold $\forall u,v,w \in V$:
  • $(u+v)+w=u+(v+w)$
  • $u+v=v+u$
• There is a zero vector $0$ such that $u+0 = 0+u = u$ for all $u \in V$ (identity)
• If $v \in V$, there exists a vector $-v \in V$ such that $v + (-v) = 0$ (inverse)
• If $u,v \in V$ then $u+v \in V$ (closure)

Note that these criteria make the vector space an abelian group. A few further conditions must be met to make it a vector space.

• Associativity and commutativity hold under scalar multiplication for constants $c_1,c_2 \in \R$:
  • $c_1(c_2v)=(c_1c_2)v$
• For all constants $c$ and vectors $v \in V$, $cv \in V$
• The distributive law holds for all constants $c, c_1, c_2$:
  • $c(u+v) = cu+cv$
  • $(c_1 + c_2)v = c_1v + c_2v$

<aside> ❗ Note that vector spaces are not restricted to vectors as an $n$-dimensional list, but anything that fits the above criteria. In fact, the set of all fixed-size matrices forms a vector space, as does the set of all polynomials! It follows that matrices and polynomials are vectors. The idea of a vector space is motivated by our thought of vectors in Euclidean $n$-space, but then the key features are abstracted to provide us with the tools to apply to more general settings - like group theory typically does.

</aside>

Subspaces

A vector space $V^\prime$ is a subspace of a vector space $V$ if $V^\prime$ is a vector space and every element of $V^\prime$ is also in $V$. There are 3 conditions we need to check.

• $\vec{0} \in V^\prime$ (identity)
• $u,w \in V^\prime \implies u+w \in V^\prime$ (addition closed)