Sample Question (pulled off Yahoo Answers): Linear Algebra Question: Finding the Dimension?

Sample Question (pulled off Yahoo Answers): Linear Algebra Question: Finding the Dimension? (Original question posted as comment on the Questions Page)

If S is a subspace in R^4 and S = {(a,b,c,d) in R^4 , a+c = b , a+c+d=0} what is the dimension of S? I have the answers and it says that S is generated by the vectors (1,0,-1,0) and (1,1,0,-1) and that the dimension of S is 2. How do we find those 2 vectors that generate S? Please explain!!

I trust that it is obvious that \( S \) is of dimension 2? This is because we can assign values to \(a\) and \(c\) essentially arbitarily and construct a unique element of \(S\).

In fact we can construct the mapping:

\[ s=(a,b,c,d)=(u,v)A \]
where: \( A= \left[ \begin{array}{cccc}1&1&0&-1\\0&1&1&-1 \end{array} \right] \), which gives us a 1-1 linear mapping from \( \mathbb{R}^2 \) to \(S\).

Hence the image of any basis of  \( \mathbb{R}^2 \) is a basis of \(S\) so in particular is:


 \( \{ (1,0)A, (0,1)A \}=\{ (1,1,0,-1),(0,1,1,-1) \} \).

One of the elements in the basis found above is in the basis you quote and the other is not. This is because the basis is not unique, any pair of linearly independent vectors in \( \mathbb{R}^2 \) will produce a basis for \(S\). The other element of your quoted basis is \( (1,-1)A=(1,0,-1,0) \).


(If you are more used to working with column vectors just take the transpose of everything above)