Question

Is the following set of Vector a basis in R3?
(1 2 3) (0 0 1) (2 2 1)
Thanks a lot!

Answer 1

What you have to verify is that the vectors are independent. If they are independent then their linear combinations will give you every possible point in three D space.

The way I like to do that is put the vectors in a matrix then see if the matrix is invertible. So in this case the matrix is:

\[\left[\begin{matrix} 1 & 0 & 2 \\ 2 & 0 & 2 \\ 3 & 1 & 1 \end{matrix}\right]\]

You go through elimination. You can see by inspection if there is a non-zero vector which, when multiplied by the matrix, gives the zero vector. You can compute the determinant (if the determinant = 0 the matrix is not invertible) - whatever you like.

If the matrix is not invertible, then the vectors are not independent and are not a basis in R3. If it is invertible, then the vectors are independent and are a basis in R3.