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]
click here for reading the complete post:
eevibes.com/ma...

[@Violet-td5bt]

Thank you so much

[@erukalaHimachandana]

But as per the condition if v1 is not equal to zero then it has to be independent??

[@eevibessite]

If you are asking about a single vector v1, then for sure

[@webcraft3315]

thank alot

[@olatunjiOlayinka970]

How did you know c3 was 1

[@yashasvithakur9078]

c3 belong to real number so we can take any value acc to our own convenience

[@jasleenkaur645]

How can you say c3 is any real number?

[@eevibessite]

@@jasleenkaur645 because that's what we suppose while writing the linear combination equation. As we are looking for some constant multiple. This contant is chosen from the set of real (so that it can be positive or a negative number )

[@jasleenkaur645]

@@eevibessite smj nhi aya mam ese to hum koi bhi number lele khud se

[@eevibessite]

@jasleenkaur645 no, the number is chosen so that the equation is satisfied. So if I have 2 vectors v1=[3 3]and v2=[6 6]
Then linear combination equation will be c1v1+c2v2=0
C1[3 3]+c2[6. 6]=0
As it can be observed easily that v2=2v1
Or if you rearrange , the equation becomes
2v1-v2=0
So here c1=2 and c2=-1

[@sagarpandey2762]

How will we choose upper triangular matrix ?

[@eevibessite]

What do you mean by choosing ? We convert the matrix into upper triangular matrix by performing elementary row operations

[@kirito1049]

Can explain the part where v1 bevame linear dependent

[@kirito1049]

Like where finding whether v1 is linear independent

[@eevibessite]

Also watch here:
th-cam.com/video/tF-WSYCZjSA/w-d-xo.html