basis for R^3

Question

{(1,2,4)^T,(2,1,3)^T,(4,-1,1)^T} is a basis for R³

Answer 1

That set is not a basis.  A basis is a set of linearly independent vectors that span the vector space.  By putting the vectors into a matrix, we can find the rank.

 

1  2  4

2  1  3

4 -1  1

 

Now we'll attempt to put this matrix into reduced row-echelon form:

 

1.  Row₃ - 2*Row₂

 

1  2  4

2  1  3

0 -3 -5

 

2.  Row₂ - 2*Row₁

 

1  2  4

0 -3 -5

0 -3 -5

 

Therefore we can see that the vectors are not linearly independent, so it is not a basis.

 

 

J.T.