Definition of linear dependence
Let v1 v2 be 2 non-zero vectors in Rn, if v1 = av2 where a is a non-zero real number, v1 and v2 are linearly dependent (or LD). If v1 =/= av2, i.e if they point to different directions, then v1 and v2 are linearly independent.
For systems of more than 2 vectors, it is more complicated.
For n vectors
In Rn, a non-zero Vector u is a linear combination of k other vectors v1, v2, … vk in Rn if it is the sum of scalar multiples of them like: u = x1v1 + x2v2 + … + xkvk.
def
A set of k vectors v1, vk are linearly independent iff not a single one of them can be written as a linear combination of the others. Alternatively they are linearly independent iff the following system (or Vector equation) x1v1 + x2v2 + … + xkvk = 0 can have only a zero solution x1 = x3 = … = xk = 0 Otherwise, they are linearly dependent.
Dimension definition
The maximum possible number of linear independent vectors in a set of k vectors if k >= n in Rn is n.
Definition of span
Let v1, v2, …, vk in Rn . The span span {v1, v2, … , vk} = {x1v1 + x2v2 … + xkvk} | x1, x2, … , xk in R}
If a Vector u (something) o in Rn is in the span {u, … vk} then u is linearly dependent on v1, v2, … vk.
If v1, v2, … vn is a set of n linearly independent vectors in Rn then Rn = span {v1, v2, … , vn}.
Let a = (-3, 1, -2), b = (-2, 0, -1). Find a Vector in span {a, b}. u = a + b = (-5, 1, -3).
Find a Vector not in the span {a, b}: v = (2, -4, 4) not in span {a, b}.
Qs
So in carthesian coordinates the dimension is dependent on the two independent vectors that we usually call x and y?