Linearly Dependent Vectors | Example of Linearly Dependent Vectors

Linearly Independent Vectors Test (Shortcut!!)
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How to test the given vectors are linearly independent or not? The vectors v1,v2,v3,...vn in a vector space V are said to be linearly dependent if there exist constants c1,c2,c3,....cn not all zero such that:

c1v1+c2v2+c3v3+......+cnvn=0 -------------------------(i)

otherwise v1,v2,v3,.....vn are called linearly independent, that is v1,v2,v3,....vn are linearly independent if whenever c1v1+c2v2+c3v3+......+cnvn=0 , we must have c1=c2=c3=0.

That is the linear combination of v1,v2,v3,....vn yields the zero vector.

How to determine either the vectors are linearly independent or not?

There are two ways to check either the vectors are linearly independent or not.

1-Graphically

2-Linear Combination Equation

Graphical Approach:

This approach is helpful for all those vectors that lie in 2D: means those vectors that have two components. Another point to consider is there should be nor more than three vectors for better understanding.

Lets consider an example:

V1=[1 2]

v2=[2 4]
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