direct sums, kernel and linear operator on a finite dimensional vector space

Question

could you please help me solve the given problem below?

Let $f$ be a linear operator on a finite-dimensional vector space $V$ over $F$.

If $$ 1+1 \ne 0 \hspace{1cm} \text{and} \hspace{1cm} f^2=2f, $$

show that $$ V=\ker(f) \oplus \ker(f-2\operatorname{id}_V). $$

I'm new to linear algebra and still struggling with direct sums, kernel and linear transformations. Tell me where to start and kindly guide me if my answer is correct. Thanks!

Answer 1

Some hints. Set $g=f-2\mathrm{id}_V$ for simplicity.

1. You want to prove that every $v\in V$ can be written in a unique way as $v=x+y$, with $x\in\ker(f)$ and $y\in\ker(g)=\ker(f-2\mathrm{id}_V)$

2. For every $v\in V$, you have $f(g(v))=0=g(f(v))$

3. For every $v\in V$, $2v=f(v)+(2v-f(v))=f(v)-g(v)$