I see that I forgot to address one point in my earlier reply.
On 29 Apr 2004 20:35:59 GMT, Anonymous wrote:
>Yes. We call all of the variables that appear first in any of the equations, >when the system is in echelon form, "basic variables". The other ones are >called "free variables". So, in any system in echelon form, the number of >basic variables equals the number of equations. Is that what you mean by >"n-r", where r is the number of free variables (dimension of the kernel)?
Yes, you understood me correctly here, but as I said in a followup, it would have been less confusing if I had chosen a different letter than r since my notational choice is probably opposite to what you will find in books. So, let's use k instead. With that change, k is the number of free variables (dimension of the kernel) and n-k is the number of basic variables.
Or, >is your "r" actually the basic variables and "n-r" the frees?
The rank of the matrix (frequently denoted r, though not in my previous posting) is the number of independent equations in the system, same as the number of basic variables, and same as the dimension of the image.
I hope you don't come away from this with the vague feeling that this is all very confusing. It's really as simple as can be. I apologize for my notational blunder; pretend that I wrote "k" in my earlier post and you shouldn't get confused.