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Cs201 Assignment No 3 Solution 2015 Tax

MTH501 assignment no 3 (due date 8 January 2017)


For a nonempty subset  of a vector space, under the inherited operations, the following are equivalent statements
  1.  is a subspace of that vector space
  2.  is closed under linear combinations of pairs of vectors: for any vectors  and scalars  the vector  is in 
  3.  is closed under linear combinations of any number of vectors: for any vectors  and scalars  the vector  is in .
Anyone have idea about it, please leave comment and  am trying  to solve

Idea About Problem no 2

S = {

 , 

}

Of vectors in the vector space R3, find a basis for span S.

The set S = {v1v2} of vectors in R3 is linearly independent if the only solution of

  c1v1 + c2v2 = 0

is c1, c2 = 0.

In this case, the set S forms a basis for span S.

Otherwise (i.e., if a solution with at least some nonzero values exists), S is linearly dependent.

If this is the case, a subset of S can be found that forms a basis for span S.

With our vectors v1v2, (*) becomes:

c1

 + 

c2

 = 

Rearranging the left hand side yields

0 c1 +1 c2

4 c1 +2 c2

1 c1 +3 c2

 = 

The matrix equation above is equivalent to the following homogeneous system of equations

0 c1

 +1 c2

=

0

4 c1

 +2 c2

=

0

1 c1

 +3 c2

=

0

We now transform the coefficient matrix of the homogeneous system above to the reduced row echelon form to determine whether the system has
  • the trivial solution only (meaning that S is linearly independent), or
  • the trivial solution as well as nontrivial ones (S is linearly dependent).

 0 

 1 

 4 

 2 

 1 

 3 

can be transformed by a sequence of elementary row operations to the matrix

 1 

 0 

 0 

 1 

 0 

 0 

The reduced row echelon form of the coefficient matrix of the homogeneous system is

 1 

 0 

 0 

 1 

 0 

 0 

which corresponds to the system

1 c1

=

0

1 c2

=

0

0

=

0

Since each column contains a leading entry (highlighted in yellow), then the system has only the trivial solution, so that the only solution of  is c1, c2 = 0.
Therefore the set S = {v1v2} is linearly independent.

Consequently, the set S forms a basis for span S and many set of solution.

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