# What's so special about the 4 fundamental subspaces?

I was reading Gilbert Strang's book for Linear Algebra (along with his lectures) and I feel that he is emphasizing that the 4 fundamental subspaces (Column Space, Row Space, Null Space and Null Space of $A^T$) form the "crux" or "heart" of Linear Algebra and that the understanding of it is crucial for further study.

But what I don't understand is WHY? Why are they so important?

I understand that :

• Row Space and Null Space are orthogonal. (Same for other two)
• Their dimensions are governed by rank of the matrix.

but beyond that I didn't find anything that I found profound or interesting. Am I missing something or is Prof. Strang over-rating the 4 fundamental subspaces?

• The short answer is that it becomes clearer the more linear algebra you learn, especially when you start applying it to do other things. I am not convinced a one-paragraph or even one-page summary of various ideas that build upon these ideas will be useful until you start tackling them yourself. Perhaps it's a good idea to simply have faith that Gilbert Strang is a very smart man who knows what he's talking about. May 29, 2012 at 20:48
• The first bullet point is not correct. May 29, 2012 at 22:11
• @QiaochuYuan. I will. Thanks ! May 30, 2012 at 15:03

If we are in this setup: $x\mapsto Ax$ for a column vector $x$ and appropriate matrix $A$, then the image of the linear transformation will be spanned by the columns of $A$.
The kernel of the transformation (nullspace) is the set of all $x$ such that $Ax=0$ is important for understanding the solutions to some matrix equations. You probably have already learned that if $x_0$ is a solution to $Ax=b$, then every other solution is given by $x_0+k$ where $k$ is in the nullspace.
This all has analogous explanation on the other side. If we are in this setup: $x\mapsto xA$ for a row vector $x$, then the image of the linear transformation is now spanned by the rows of $A$.
Talking about the nullspace of $A^T$ is just a fancy way of dressing up the "left nullspace" of $A$, since $xA=0$ iff $A^T x^T=0$. The nullspace is now the set of all $x$ such that $xA=0$, and you can draw the same conclusions about solutions to $xA=b$.
In short, these four spaces (really just two spaces, with a left and a right version of the pair) carry all the information about the image and kernel of the linear transformation that $A$ is affecting, whether you are using it on the right or on the left.