Give an example of a linear transformation $T:\mathbb R^2 \to \mathbb R^2$ such that $N(T)=R(T)$.

Solution 1:

Let

$\mathbf e_1, \; \mathbf e_2 \in \Bbb R^2 \tag 1$

be a basis; define

$T: \Bbb R^2 \to \Bbb R^2 \tag 2$

on this basis by

$T \mathbf e_1 = \mathbf e_2, \tag 3$

$T\mathbf e_2 = 0, \tag 4$

and extend $T$ to all of $\Bbb R^2$ by linearity:

$T(a \mathbf e_1 + b \mathbf e_2) = a T\mathbf e_1 + b T\mathbf e_2 = a\mathbf e_2; \tag 5$

then it is easy to see that

$T(a \mathbf e_1 + b \mathbf e_2) = 0 \Longleftrightarrow a = 0, \tag 6$

whence

$N(T) = \{b \mathbf e_2, \; b \in \Bbb R \}, \tag 7$

and also from (5)

$R(T) = \{a \mathbf e_2, \; a \in \Bbb R\}. \tag 8$

It is now trivially obvious to even the most casual observer that

$N(T) = R(T). \tag 9$