Linear combination and vector equations in $R^n$  Linear Equations with Matrices
Linear combination and vector equations in $R^n$
Lessons
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{$v_1,\cdots,v_p$}.

Intro Lesson
Vector Equations in $\Bbb{R}^n$ Overview:

1.
Calculating Vectors in $\Bbb{R}^n$
Consider the two vectors , and . Compute: 
4.
Linear Combinations with Known terms
Determine if $b$ is a linear combination of $a_1$, $a_2$ in part a. Determine if $b$ is a linear combination of $a_1$, $a_2$, and $a_3$ in part b and c.