Finding a Different Basis for Echelon Form

In summary, the problem is to find another matrix with the same echelon form but different basis, but the question is ambiguous. It is not clear what is meant by "basis" and whether the entries in the matrix can be modified. The given matrix is not in row echelon form, so it is not clear how it is related to the problem.
  • #1
EvLer
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0
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
 
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  • #2
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
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:

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

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).
 
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  • #4
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
EvLer said:
but when they say "the same row echelon form" doesn't it mean i cannot modify the matrix entries?
thank you.

What do you mean? To get it into http://en.wikipedia.org/wiki/Row_echelon_form" , but that matrix isn't in that form.
 
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FAQ: Finding a Different Basis for Echelon Form

1. What is a basis for echelon form?

A basis for echelon form is a set of linearly independent vectors that span the same vector space as the original set of vectors. It can also be described as a set of vectors that form the columns of the matrix in echelon form.

2. Why is finding a different basis for echelon form important?

Finding a different basis for echelon form can be important for a few reasons. Firstly, it can help simplify calculations and make solving linear systems easier. Additionally, it can provide a different perspective on the vectors and their relationships, which can be useful in understanding and solving problems.

3. How do you find a different basis for echelon form?

To find a different basis for echelon form, you can use the process of Gaussian elimination. This involves performing row operations on the original set of vectors to reduce them to echelon form. The resulting columns will then form a different basis for the vector space.

4. Can a different basis for echelon form be unique?

Yes, a different basis for echelon form can be unique. This means that there is only one set of vectors that can form the columns of the matrix in echelon form. However, there can also be multiple sets of vectors that can form a different basis for echelon form.

5. Is finding a different basis for echelon form always possible?

No, finding a different basis for echelon form is not always possible. This is because not all sets of vectors can be reduced to echelon form using Gaussian elimination. In these cases, the original set of vectors is said to be linearly dependent and a different basis for echelon form cannot be found.

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