Vector Addition
Given two vectors u and v in Rn $$ u=(a_1,a_2,...,a_n) \\ u=(b_1,b_2,...,b_n) $$ the sum u + v p is a vector in Rn obtained by adding the corresponding components of u and v $$ u+v=(a_1+b_1,a_2+b_2,...,a_n+b_n) $$
The sum of vectors with a different number of components is not defined.
Example
Given two row vectors in R3
$$ u=(3,4,1) $$
$$ v=(1,-1,1) $$
The sum of the vectors u + v is another vector in R3
$$ u+v=(3+1,4+(-1),1+1) = (4,3,2) $$
Example
Given two column vectors in R3
$$ u= \begin{pmatrix} 3 \\ 4 \\ 1 \end{pmatrix} $$
$$ v=\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} $$
The sum of the vectors u + v is another vector in R3
$$ u+v=\begin{pmatrix} 3 \\ 4 \\ 1 \end{pmatrix} + \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3+1 \\ 4+(-1) \\ 1+1 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \\ 2 \end{pmatrix} $$
Theorems
- Vector addition satisfies the associative law for addition. Given three vectors u, v, w $$ (u+v)+w=u+(v+w) $$
- Vector addition satisfies the commutative law for addition. Given two vectors u, v $$ u+v=v+u $$
- Let v be a vector and o be the null vector in Rn, the addition v + o is equal to the vector v. $$ v+o=v $$
- Let v be a vector and -v be the negative vector of v in Rn, the addition v + (-v) is equal to the null vector o. $$ v+(-v)=o $$