A Kernel Of A Matrix: An In-Depth Guide To The Concept.

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The concept of kernel is quite often explained with the help of a matrix. While it can be a great reference tool, it doesn’t always give an accurate depiction of the concept. In this article, we will learn about kernels using a more detailed and practical example. The problem with Matrices is that they usually involve too many concepts at once. That makes them hard to understand for beginners. In this guide, we will try and explain the concept in a detailed manner that is easy to understand without losing any of the important details.

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Kernal(A) is nothing but it is the (non-trivial) solution space of Ax=0. (This Ax=0 is called the homogenous system of equations.) That is, there are infinite number of non-trivial solutions of Ax=0. Collectively they (infinite solutions) are called Kernal of A, i.e. ker(A).
NB: trivial solution of Ax=0 is a unique known solution, x=0.

For Ax=b (non-homogeneous system of equations, where b is non-zero vector), if there exists kernal(A). (It means there are non-trivial solutions for Ax=0.) In this case, collectively these infinite number of solutions is also called null space (i.e. kernel(A)) of Ax=b. This is the case if Ax=b is consistent non-homogenous system of equations. For inconsistent Ax=b there is no solution of x at all in theory!

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