# Column space

1. Sep 20, 2006

### EvLer

Here's the problem:
given in echelon form, the column space basis is [5,0,0,0]^t, [4,2,0,0]^t
and the question is to find another matrix A with the same echelon form but different basis....
how do i find a different basis?

thanks

2. Sep 21, 2006

### EvLer

if I just reduce the basis like this:
[1,0,0,0]^t and [2,1,0,0]^t would that count as a new basis with the same echelon form?
could someone confirm please? this problem is very ambiguous to me, i can find another basis but the echelon form will change!

3. Sep 21, 2006

### StatusX

You'll have to explain the problem a little more clearly. I've only heard of (row) echelon form as regarding matrices. Are you given the matrix:

$$\left[ \begin{array}{cc} 5 & 4 \\ 0 & 2 \\ 0 & 0 \\ 0 & 0 \end{array} \right]$$

which is in row echelon form, and asked to different matrix with the same row echelon form? (I don't know what you could mean by basis, other than the basis of the column space given by the columns of the matrix) This isn't a good question, as the row echelon form is not unique. If you want to find another matrix which can be put in that form by gaussian elimination, just change any of the terms on or above the diagonal (obviously not making diagonal terms zero).

Last edited: Sep 21, 2006
4. Sep 21, 2006

### EvLer

0 5 4 3
0 0 2 1
0 0 0 0
0 0 0 0
is the matrix and yes, the basis is the basis of the col space... but when they say "the same row echelon form" doesn't it mean i cannot modify the matrix entries?
thank you.

5. Sep 21, 2006

### StatusX

What do you mean? To get it into row echelon form, you need to use elementary matrix transformations like multiplying rows by constants and adding two rows, and these can be applied to a matrix in row echelon form to get another matrix in row echelon form, and in this sense the form isn't unique. Uniqueness does apply to reduced row echelon form, but that matrix isn't in that form.

Last edited: Sep 21, 2006