To find the kernel (null space) of the linear transformation T:
, defined by
, where
, we need to solve the equation
.
Using
tags for formatting, the matrix equation can be represented as:

Substituting the values of
, we have:

To find the kernel, we need to solve the augmented matrix
using row reduction techniques:

Performing row reduction, we get:

Dividing the first row by 10 and the second row by -2, we have:

From the row-reduced form, we can see that
and
are leading variables, while
is a free variable.
Therefore, the kernel (null space) of the transformation T is given by:

In set notation, we can represent the kernel as:

Therefore, the correct option is:
.