A matrix with one column is called a

**column vector**. They can be added or subtracted with other column vectors as long as they have the same amount of rows.

**Parallelogram Rule for Addition:** if you have two vectors

$u$ and

$v$, then

$u+v$ would be the fourth vertex of a parallelogram whose other vertices are

$u,(0,0)$,and

$v$
Here are the following algebraic properties of

$\Bbb{R}^n$
1.

$u+v=v+u$
2.

$(u+v)+w=u+(v+w)$
3.

$u+0=0+u=u$
4.

$u+(-u)=-u+u=0$
5.

$c(u+v)=cu+cv$
6.

$(c+d)u=cu+du$
7.

$c(du)=(cd)(u)$
8.

$1u=u$
Given vectors

$v_1,\cdots,v_p$ in

$\Bbb{R}^n$ with scalars

$c_1,\cdots,c_p$, the vector

$x$ is defined by

$x=v_1 c_1+\cdots+v_p c_p$
Where

$x$ is a linear combination of

$v_1,\cdots,v_p$.

The linear combinations of

$v_1,\cdots,v_p$ is the same as saying

**Span{$v_1,\cdots,v_p$}**.