# Visualizing the four subspaces of a matrix

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?

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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

## 1 Answer

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

Here is the original source.

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

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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