A How Does the Non-Equality of Kernels Imply Their Sum Equals the Vector Space?

[Portuga]

TL;DR

Let $F$ and $G$ be two non-zero linear functionals over a vector space $V$ of dimension $n$. Assuming $ker (F ) \neq \ker (G)$, determine the dimensions of the following subspaces: $\ker (F )$, $\ker (G)$, $\ker (F ) + \ker (G)$, and $\ker (F ) \cap \ker (G)$.

This is actually a solved exercise from a Brazilian book on Linear Algebra. The author presented the following solution:

The kernel and image theorem tells us that dimension $\dim V=n=\dim\ker\left(F\right)+\dim \text{im}\left(F\right)=\dim\ker\left(G\right)+\dim\text{im}\left(G\right)$. As $\text{im}\left(F\right)\subset R$, $\dim\mathbb{R}=1$ and $F\neq0$, then $\dim\text{im}\left(F\right)=1$. Similarly $\dim\text{im}\left(G\right)=1$. Therefore, $\dim\ker\left(F\right)=\dim\ker\left(G\right)=n-1$. On the other hand, the dimension theorem of the sum assures us that

$$
\dim\left(\ker\left(F\right)+\ker\left(G\right)\right)+\dim\left(\ker\left(F\right)\cap\ker\left(G\right)\right)=\dim\ker\left(F\right)+\dim\ker\left(G\right)=2n-2.
$$

In general, $\ker\left(G\right)\subset\ker\left(F\right)+\ker\left(G\right)$ and due to the hypothesis $\ker\left(F\right)\neq\ker\left(G\right)$, we will have $\ker\left(F\right)\begin{array}{c} \subset \\ \neq \end{array}\ker\left(F\right)+\ker\left(G\right)$; then necessarily $\ker\left(F\right)+\ker\left(G\right)=V$. So $\dim\left(\ker\left(F\right)+\ker\left(G\right)\right)=n$ and hence

$$
\dim\left(\ker\left(F\right)\cap\ker\left(G\right)\right)=\left(2n-2\right)=n-2.
$$

I am ok with almost everything he presented, but couldn't understand why

the hypothesis $\ker\left(F\right)\neq\ker\left(G\right)$ implies that $\ker\left(F\right)+\ker\left(G\right)=V.$

Any ideas?

Thanks in advance.

 

[Office_Shredder]

Do you understand why $ker(F)\subsetneq ker(F)+ker(G)$?

Since the left hand side has dimension n-1, the right hand side must have dimension n or higher.

 

[Portuga]

Do you understand why $ker(F)\subsetneq ker(F)+ker(G)$?

Since the left hand side has dimension n-1, the right hand side must have dimension n or higher.

Oh my god! It was so simple!
Thank you very much!