Why is Understanding Column Space and Null Space Important in Linear Algebra?

[Muthumanimaran]

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

 

[economicsnerd]

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 $A$

 

[Muthumanimaran]

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.