Why Linear Dependence/Independence Formulas Make Sense
To understand linear dependence and linear independence we first need to understand the linear combination of two or more vectors
Linear Combination
A linear combination of two or more vectors is an expression which is constructed from a set of vectors in which each vector is multiplied by a scalar and then adding the results.
a1v1 + a2v2 + a3v3 + ….. + anvn
where v1,v2,…,vn are vectors and an are scalars which scales the corresponding vectors.
Lets consider a 2D plane in which we have two vectors [1,2] and [4,5]. By the linear combination of these two vectors we can reach any point on the 2D plane.
Span
The span of two vectors v1 and v2 is the set of all their linear combinations.
OR
The set of all possible vectors which can be reach through the linear combination of two vectors V1 and V2 is the span of those two vectors.
What Is The Span Of A Single Vector
The span of a single vector is all the vectors which lie on the single line.
Linear Dependence
Lets say we have two vectors in a 2D plane and they are collinear that is one of the vector is redundant. It means one of the vector is not adding anything to the span of the first vector. In such case the two vectors are known as linearly dependent.
Mathematical Definition of Linear Dependence
Let S be the set of vectors
S = {V1, V2, V3,…..,Vn}
The set S is linearly dependent if and only if CV1+ C2V2 + C3V3 +….+ CnVn=zero vector
for some all Ci’s at least one is non zero.
The condition of checking linear dependence
if c1 or c2 is non zero then the two vectors are linearly dependent
Linearly Independence
If in a 2D plane the two vectors V1 and V2 are not collinear then one of the vector is increasing the span of the first vector that is with only vector the span was just a single line but with the linear combination of V1 and V2 we can reach every single vector in the 2D plane(Span of V1 and V2 is the whole 2D plane). It means that no vector is redundant. In such case the two vectors are known as linearly independent.
Mathematical Definition of Linear Independence
Let S be the set of vectors
S = {V1, V2, V3,…..,Vn}
The set S is linearly independence if and only if CV1+ C2V2 + C3V3 +….+ CnVn=zero vector
The condition of checking linear independence
if c1 and c2 are both zero then the two vectors are linearly independent
In our example since c1 and c2 are both zero, therefore V1 and V2 are linearly independent.
But Why This Formula Makes Sense?
The conditions to check the linear dependence/independence basically check whether the two vectors in the 2D plane are collinear or not. Lets dive into it deeper.
We know that to find the linear combination of two vectors we multiply the vectors by some scalar and add them. Since we equated our linear combination of V1 and V2 to zero vector .It means we are basically asking the question that to reach the zero vector by the linear combination of V1 and V2 by which scalar we need to multiply our vectors. We got c1=c2=0 in our example that means the only way to reach the zero vector by the linear combination of V1 and V2 is to multiply those vectors by 0. This shows that the two vectors V1 and V2 do not lie on the same line and hence they are Linearly independent because the only way to reach the zero vector by the linear combination of V1 and V2 is to scale both the vectors by zero.
Note:
If V1 and V2 were collinear there will be infinite values of c1 and c2 through which we can reach the zero vector by the linear combination of two vectors.(The vectors will be opposite and direction having the same magnitude).
I assumed that we are working in 2D plane. The concept of Linear Dependence/Independence also applies to higher dimensions but this intuition of collinearity will not be applicable in higher dimensions.
If you find any mistake or there is something you didn’t understand then kindly comment. I would love to hear.
Thanks for reading :)
