vector space endomorphism with kernel as image and dimension

Proposition

Let \(\varphi: V \rightarrow V\) be a vectorspace-endomorphism between finite dimensional vector spaces with

\begin{align*} \mathrm{ker}(\varphi) = \mathrm{im}(\varphi) =: U \end{align*}

Then

\begin{align*} 2\cdot \mathrm{dim}(U) = \mathrm{dim}(V) \end{align*}

Proof

choose a basis \(u_1,...,u_n\) of \(U\).
Then since \(u_1,...,u_n \in \mathrm{im}(\varphi)\) we may choose preimages \(w_1,...,w_n\), i.e. \(\varphi(w_i) = u_i\).
We claim that \(\{u_1,...,u_n,w_1,...,w_n\}\) forms a basis.
If we can show this, then we are done as this shows \(\mathrm{dim}(U) = n, \mathrm{dim}(V) = 2n\).

linearly independent

Suppose we have a linear combination \(\sum_{i=1}^{n} \lambda_i u_i + \sum_{j=1}^n \lambda'_j w_j = 0\).
Then we may apply \(\varphi\) and get

\begin{align*} 0 =& \varphi(0) \\ =& \varphi(\sum_{i=1}^{n} \lambda_i u_i + \sum_{j=1}^n \lambda'_j w_j)\\ =& ... \\ =& \sum_{i=1}^n \lambda_i \varphi(u_i) + \sum_{j=1}^n \lambda_j' \varphi(w_j) =& \sum_{j=1}^n \lambda_j' u_j \end{align*}

since \(u_i \in U =\mathrm{ker}(\varphi)\).

Now we use that \(u_j\) is linearly independent to conclude that \(\lambda'_j = 0\) for \(j \in \{1,...,n\}\).
Hence we get \(\sum_{i=1}^n \lambda_i u_i = 0\) and again (since \(u_i\) is linearly independent) \(\lambda_i = 0\).

generates \(V\)

Let \(v \in V\) be arbitrary.
Then \(\varphi(v) \in \mathrm{im}(\varphi)\), so in particular \(\varphi(v) = \sum_{i=1}^n \lambda_i u_i = \sum_{i=1}^n \lambda_i \varphi(w_i) = \varphi(\sum_{i=1}^n \lambda_i w_i)\).
Now consider \(v - \sum_{i=1}^n \lambda_i w_i\).
It lies in the kernel since \(\varphi(v - \sum_{i=1}^n \lambda_i w_i) = \varphi(v) - \varphi(\sum_{i=1}^n \lambda_i w_i) = 0\) and thus \(v - \sum_{i=1}^n \lambda_i w_i = \sum_{j=1}^n \lambda_j' u_i\).
Then putting both results together yields

\begin{align*} v =& v - \sum_{i=1}^n \lambda_i w_i + \sum_{i=1}^n \lambda_i w_i \\ =& \sum_{j=1}^n \lambda_j' u_i + \sum_{i=1}^n \lambda_i w_i \end{align*}

which shows that \(v \in \langle u_1,...,u_n,w_1,...,w_n\rangle\)