Is there a linear transformation from $\mathbb R^2\to\mathbb R^2$ with $\dim(\ker T)=\dim(\mathrm{im}T)=1$ and $\mathbb R^2=\ker T\oplus\mathrm{im}T$?

Question

Is there a linear transformation from $\mathbb R^2$ to $\mathbb R^2$ that: $$\dim (\ker T) = \dim (\operatorname{im} T) = 1$$

and also

$$\mathbb R^2 = \ker T \oplus \operatorname{im} T?$$

Thank you.

Answer 1

Sure, take $T\colon ℝ^2 → ℝ^2,~(x,y) ↦ (x,0)$. Now what is $\ker T$ and $\operatorname{img} T$?

An important class of examples of such transformations $T$ on a vector space $X$ with the property $V = \ker T \oplus T$ are so-called projectors, see wiki/Projector. These are endomorphisms $p$ on a vector space $V$ such that $p^2 = p$. It’s easy to see that, for these, $\ker p ∩ \operatorname{img} p = 0$ and $\ker p + \operatorname{img} p = V$.