Linear combination and vector equations in $R^n$ - Linear Equations with Matrices

Notes:

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{$v_1,\cdots,v_p$}.