$R^n$

The set of all vectors with n entries is denoted by $R^n$ . The R stands for the real numbers that appear as entries in the vectors, and the exponent n indicates that each vector contains n entries

Geometric Description of $R^2$

we can identify a geometric point $(a,b)$ with the column vector

$$ \begin{bmatrix}a\\b\end{bmatrix} $$

so we may regard $R^2$ as the set of all points in the plane

Linear Combinations

Given vectors $v_1 , v_2 , ... v_p$ in $R^n$ and given scalars $c1,c2, ... cp$ the vector $y$ defined by

$$ y = c_1v_1 , c_2v_2 , ... c_nv_n $$

is called a linear combination of $v_1 , v_2 , ... v_p$ with weights $c1,c2, ... cp$

Visualization

$$ v_1 = \begin{bmatrix}-1\\ 1\end{bmatrix} \\ \\ v_2 = \begin{bmatrix}2\\ 1\end{bmatrix} $$

linear combination of v1 and v2 with integer weights