Simpler definition for kernel, image, and rank of transformation matrix?

Question

I'm a beginner at linear algebra and I keep running into such complicated definitions for kernel, image, and rank of a matrix and I would appreciate it if someone could simplify the definitions more into English rather than math symbols. I generally know how to find these qualities but...what am I finding? Thanks in advance!

Answer 1

Viewing a matrix as a function (given by multiplying by the matrix), the definitions translate to:

• The kernel is everything that is mapped to zero by the function.
• The image is everything that comes out of the function.
• The rank is the dimension of the image.

For the latter, you should first verify that the image of a matrix is a vector space.

Answer 2

Without symbols, the definitions become unnecessarily lengthy! But, per request:

(i) The kernel of a linear map is the set of all points of the domain of the linear map such that the linear map assigns the zero of its codomain to each of the points. (cf. The kernel of a linear map $T$ is the set of all points $x$ of the domain of $T$ such that $T(x) = 0$).

(ii) The image (range, more properly, used hereafter) of a linear map is the set of all points of the codomain of the linear map such that the linear map assigns each of the points to at least one point of the domain of the linear map. (cf. The range of a linear map $T$ is the set of all points $y$ of the codomain of $T$ such that $y = T(x)$ for some $x$ in the domain of $T$.)

(iii) The rank of a linear map is the dimension of the range of the linear map.

Of course we can simplify (iii) by assigning a symbol to the range of $T$. Moreover, using the language of set theory we can further simplify (i) and (ii); for example: let $V, W$ be vector spaces; let $T: V \to W$ be linear. Then the kernel of $T$ is defined as the set $\{ x \in V \mid T(x) = 0_{W} \}$.