I am confused. I am asked to give the basis of a column space, row space, null space and even orthonormal bases. All fine, yet I know that we have either a RREF (reduced row echelon form), where the leading entries are 1's and everything else in that same column is a 0 and REF where it's not essential that the other numbers are 0's and can be anything, as long as the pivots are 1's.
Well, my question now is: for which should I go?
Let's say I have reduced this matrix X:
1 0 0 2
0 1 5 2
0 0 1 2
0 0 0 0
I have made the leading entries bold. This should be a REF, because we can clearly see in x3 that we have a 5 above the 1, which doesn't make it RREF anymore.
My question is: would that make a difference for the calculations of the things I summed up above?
Or let me put it this way: when should I explicitly go for a RREF?
Best Answer
Both forms are fine for the row space: as soon as you're in row-echelon form, you know that the remaining nonzero rows are linearly independent, so they form a basis for the row space.
For the column space, you're done as soon as you find out which columns will contain the pivots (which you also already know from the row-echelon form). Then the corresponding columns of the original matrix form a basis for the column space.
It's for the null space that you may want a reduced row-echelon form: finding a basis for the null space amounts to solving for the pivot variables in terms of the free variables. So you either go for the reduced row-echelon form, and then you can read off the expressions you want directly, or you stop at the row-echelon form and do a lot of back-substitution.
Finally, for finding an orthonormal basis, you use the Gram-Schmidt algorithm, which is not row reduction to begin with.