Hour 14,15 & 16 - Image and Kernel

Hello Everyone! Welcome back again to the concept blog and today we’re gonna cover something really interesting: Images and Kernel! For any matrix A, the kernel and image are subsets in it’s domain and co-domain which are really useful properties of the matrix A!

Hour 14

Kernel

The kernel (or null space) of a matrix A, written ker(A), is the set of vectors x with Ax = 0. So, when you multiply the matrix A and any vector x in the ker(A), the result must be the zero vector.

Note that kernel is a subspace and it holds all the properties of a subspace. This means it has a basis. To find the basis of the kernel of a matrix A, we have to solve for matrix A by making it a homogenous system (the outputs must be all zero).

So, the solutions is the basis of the kernel because any vector from the kernel would output the zero vector.

Image

The image (or column space) of a matrix A, written im(A), is the set of vectors y with y = Ax for some x. So, it is basically all the possible outputs of the function and you can consider this as the range too. It is basically all the possible values of y that can exist, given a valid input vector x such that Ax = y.

Just like the kernel, the image is also a subspace and we can also find the basis of the image. To find the basis of Im(A), we have to take the linear combination of all the linearly independent columns of the matrix A, that is the span of the linearly independent columns of the matrix A.

Which Dimension does the Image and the Kernel belong to?

For an m x n matrix A, the Ker(A) must belong to Rn because any vector x in the kernel must have rows equal to the column of the matrix A, in order for Ax to be valid. Similarly, since the matrix A is m x n, the output Ax must be in Rm.

## Hour 15

Now that we learnt about what Image and Kernel are, let’s recall some important concepts which would be really helpful to understand the Rank-Nullity Theorem.

## Basis

The basis of a subspace V is defined as the minimal set of independent vectors needed to span V. For a subspace V, there can be more than 1 basis as long as the vectors in the basis are linearly independent to each other and span V.

## Dimensions

The dimension of a subspace V is simply the number of vectors in the basis of V. The basis of the image of a matrix A is the linearly independent columns and so the rank of the matrix A is actually the dimension of the Im(A).

## Subspace

Finally, we all know by now what subspaces are and we must be aware of the properties of subspaces because these also hold true for Images and Kernels because they are also subspaces!

Hour 16

The Rank-Nullity Theorem

So far, we have seen what an Image and a Kernel are. But is there any link between the two? Turns out there is a very crucial link and this is described by the Rank-Nullity Theorem.

The Rank-Nullity theorem states that for a matrix A, the dimension of the Im(A) = Rank(A) and the dimension of Ker(A) = Nullity(A). Rank(A) is the number of leading one’s in the matrix A, and the Nullity(A) is the number of free variables in the matrix A. Note that, Rank(A) + Nullity(A) must be equal to the number of columns of A.

From this we can conclude the wonderful relationship that the Dim(Im(A)) + Dim(Ker(A)) = No. of columns of A.

So, this was the Rank-Nullity theorem and we can use this theorem to find the dimensions of Kernels and Images, given any one of them.

So, that’s all for today! See you in the next blog!