If the matrix isn't square, then you have to find the Ker and Im every time without short cuts when transposed.
I'm pretty sure there have to be some easier ways of finding bases for those two spaces other than taking the transpose of a given matrix A and row reducing it again (after row reducing the original A to find Im(A) and Ker(A)). Does anyone know? help would be appreciated. thanks!
the matrix A was this:
[1 1 0 1
1 0 3 -1
1 0 1 -1
1 2 0 3]
I found a basis for Im(A) to be the first 3 columns of A and a basis for Ker(A) to be the vector [-3 0 2 3].
I now have to find the orthogonal projections onto the kernal and image of A-transposed
another question, not related to my original one:
it says to find an orthonormal basis for Im(A)
would you just use the gram-schmidt process? I'm given a hint that points out that two vectors of the basis I found for Im(A) are already orthogonal but I don't know what to do with it