# Column space and null space

Why it is important to know about Column space and Null spaces in Linear Algebra?

## Answers and Replies

Related Linear and Abstract Algebra News on Phys.org
If ##T: X\to Y## is a linear map between vector spaces, then there are a bunch of different reasons to care about the kernel ##\text{ker}T = \{x\in X:\enspace Tx=0\} \subseteq X## and range ##\text{ran}T = \{Tx: \enspace x \in X\} \subseteq Y##. Why/whether we care about those depends on why we care about the map ##T##.

In the special case where ##X## and ##Y## are Euclidean and ##T## is represented by a matrix ##A##, the kernel of ##T## is exactly the null space of ##A##, while the range of ##T## is exactly the column space of ##A##

Thank you. But I have not done linear mappings yet. I am reading Linear Algebra and its applications by Gilbert strang, 4th edition. while I am reading subspaces (chapter 2) I was wondering what is the use of such subspaces. If you can explain me intuitively without linear mapping it would be very helpful.