# Math Help - sequence and basis of subspace!! help!

1. ## sequence and basis of subspace!! help!

In the real vector space R^4, find a basis for the subspace Span(S), where S is
the sequence
(1, 0, 2, 0), (2, 2,−2, 4), (1, 1,−1, 2), (0, 2,−6, 4), (2, 0, 4, 0).

(0 ,1 ,-3 ,2).
this method i can see they wrote the vectors in rows and do suitable row operations.
But i used the method which taught in the lecture ,and i got the basis is (1,0,2,0),(1,1,-2,2),(0,2,-6,4).
(i wrote the vectors in columns and did row operations)

which one is right? can the subspace Span(S) have different basis??? Thank you very much!!

2. First of all, obviously there are inifinite possibilities for bases in every vector space. You can always multiply all of a given basis elements in a scalar and get a new, different, legitimate basis, for example.
However, the dimension of a vector space is defined well. consequently, you and your source cannot both be right, since you found 3 linearly independent vectors which form (in your opinion) a basis, meaning that dim(V) = 3, while according to your source, dim(V) = 2.
Sorry to say that your source is indeed correct. The vector space is spanned by these two vectors: {(1,0,2,0),(0,1,-3,2)}.
That means that each vector, including the others you've written, can be given as a linear combination of these two vectors.
To find that out - write all of the vectors as columns (or rows) in a matrix, and use Gaussian elimination to find the two linear independent vectors I've mentioned.
(Notice that there are other possibilities, but these two are the ones I got as well by elementry actions, and I think so should you.)

3. Im clear with this now. i figured out the mistake i made, because i copied one wrong vector from the question. but anyway thank you very much.