Linear Dependence Rank Method 3





In this page linear dependence rank method 3 we are going to see some example problem to understand how to test whether the given vectors are linear dependent.

Procedure for  Method II

  • First we have to write the given vectors as row vectors in the form of matrix.
  • Next we have to use elementary row operations on this matrix in which all the element in the nth column below the nth element are zero.
  • The row which is having every element zero should be below the non zero row.
  • Now we have to count the number of non zero vectors in the reduced form. If number of non zero vectors = number of given vectors,then we can decide that the vectors are linearly independent. Otherwise we can say it is linearly dependent.

Example 3:

Test whether the vectors (1, 3, 1), (-1, 1, 1) and (2, 6, 2) are linearly dependent.

Solution:

linear dependence rank method 3

˜
 
1 3 1
-1 1 1
2 6 2
 

R₂ => R₂ + R₁

        -1         1       1

         1        3        1

      _________________

       0         4       2

      _________________

R₃ => R₃ - 2R₁

         2         6       2

         2        6        2

        (-)      (-)     (-)

      ___________________

       0       0       0

      ___________________

linear dependence rank method 3  linear dependence rank method 3


˜
 
1 3 1
0 4 2
0 0 0
 


R₂ => R₂ + R₁

R₃ => R₃ - 2R₁

Number of non zero rows is 2. So rank of the given matrix = 2.

If number of non zero vectors = number of given vectors,then we can decide that the vectors are linearly independent. Otherwise we can say it is linearly dependent.

Here rank of the given matrix is 2 which is less than the number of given vectors.So that we can decide the given vectors are linearly dependent.

Related pages

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.” linear dependence3 rank method





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