Given two vectors x,y where length of x = 1, length of y = 5 and their scalar product = 3. How could I show the ordered pair {x,y} is a basis?
Suppose that they are dependent, so there exists $\displaystyle \alpha \in \mathbb{R}$ such that
$\displaystyle X=\alpha Y$
put $\displaystyle Y=(y_1,y_2)$, then $\displaystyle X=(\alpha y_1,\alpha y_2)$
Now,
$\displaystyle 3=\left< X,Y\right>=\alpha (y_1^2+y_2^2)=\alpha \Vert Y \Vert ^2=25\alpha $
hence $\displaystyle \alpha =\frac{3}{25}$, thus
$\displaystyle \Vert X \Vert=\sqrt{(\alpha y_1)^2+(\alpha y_2)^2}=\alpha \Vert Y \Vert =\frac{3}{5}$
a contradiction. Therefore, they are independent