Question

How can the linear combination of two non singular and linearly independent vectors encompass the whole of 2D euclidean space?

Answer 1

Hoot! You just asked your first question! Hang tight while I find people to answer it for you. You can thank people who give you good answers by clicking the 'Good Answer' button on the right!

Answer 2

You asked how so I am assuming the question isn't why you need specifically, 2 linearly independent vectors. Well consider the independent vectors v1 and v2 being (a,0) and (0,b) . I can get any vector (c,d) by multiplying v1 and v2 by some 2 scalar values and adding v1 and v2 together:\[\alpha _{1}(a,0)+\alpha _{2}(0,b) = (\alpha _{1}a, \alpha _{2}b)=(c,d)\]
The simplest case to consider is the elementary vectors above with a and b both equaling 1. Then you just need to specify two alpha values.