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Excercise 7-2.
Find a basis for the subspace spanned by the following set:
Once you have found the basis, state the dimension of the subspace it spans and write the dependent vectors in terms of the independent vectors.
Row reduce the following matrix to find the pivotal columns:
With the matrix in RREF we can see that
so the subspace is a plane in
. The basis of this plane is made of the linearly independent vectors
and
:
The dependent vectors can be written in terms of the independent vectors based on the numbers in the non-pivotal columns:
Figure
7-1
shows the two basis vectors and the linear combinations of basis vectors that give
and
.
Object could not be loaded.
Figure 7-1. Basis Vectors
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Once you have found the basis, state the dimension of the subspace it spans and write the dependent vectors in terms of the independent vectors. 
so the subspace is a plane in 
. The basis of this plane is made of the linearly independent vectors 
and 
:
and 
.