Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given a system of linear equations in the form $$AX=b$$ How can I go about visualizing the four fundamental sub-spaces - column space, row space, null space and left null space?

In the same context, how can I visualize the orthogonality of row space and null space, and column space and the left null space?

share|cite|improve this question
Before you imagine the subspaces, how do you imagine $n$ dimensions? – Calvin Lin Jan 7 '13 at 8:20
I don't know many people who can visualize more than 3 dimensions, but as far as this problem is concerned, I would be more than happy if I could just visualize the 4 subspaces in 3 dimensions1! – Chethan Ravindranath Jan 7 '13 at 8:23
Einstein could do that, but... ;-) – Babak S. Jan 7 '13 at 8:29
Since "subspace of a matrix" is not really a standard expression, I have to ask this. Do "the four subspaces" refer to left/right nullspace, columnspace and rowspace? – rschwieb Jan 7 '13 at 14:26
Yep! I edited the question to be more clear. Thanks! – Chethan Ravindranath Jan 8 '13 at 3:29
up vote 3 down vote accepted

I could type it all out, but I think this most efficiently gets you toward what you are after.

enter image description here

Here is the original source.

Here is another way to think about these things from Gilbert Strang:

enter image description here

share|cite|improve this answer
Thanks JohnD! I am still trying to comprehend the first explanation. I am taking the course by Gilbert Strang and have come across the second picture. Hopefully, I will be able to get a clear picture from your answer! – Chethan Ravindranath Jan 9 '13 at 14:52
What does the null space of C(A) look like? Is it inside the Columnspace of all Ax? – in code veritas Dec 27 '14 at 17:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.