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Determine which of the following sets of vectors form a basis for R 3 . S...

Determine which of the following sets of vectors form a basis for R 3 . S = {(1, 0, −1),(2, 5, 1),(0, −4, 3)}, T = {(−1, 3, 2),(3, −1, −3),(1, 5, 1)}.

Solutions

Expert Solution

Solution:

Check whether is linearly independent.

Let   

This in matrix form is

By  

By  

is linearly independent.

Check  .

Let .

To find   such that  .

This in matrix form is

By  

By  

  .

.

Hence,   is the basis of .

--------------------------------------------------------------------------------------------------------------------------------

Check whether   is linearly independent.

Let   

This in matrix form is

By  

By   

Since the rank of the matrix is less than the number of unknown , there is   free variable.

Since   are not zero ,   is linearly dependent and hence not a basis of .

milcah answered 1 year ago
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