TITLE: non zero linear functional and which of the following statements are true. (NBHM-$2014$)
QUESTION [5 upvotes]: Let $V$ be finite dimensional real vector space and let $f$ and $g$ be non zero linear functionals on $V$.  Assume that $\ker(f) \subset \ker(g).$ Which of the following are true??
a. $\ker(f)=\ker(g)$
b. $f=\lambda g$ for some real number $\lambda \ne 0$.
c. The linear map $A\colon V\to \mathbb{R}^2$ defined by $Ax=(f(x),g(x))$ for all $x \in V$, is onto.
Since $\ker(f)\subset \ker(g)$ we get $\ker(f)=\ker(g)$ and hence (a) and (b) are true. Now the linear map will look like $Ax=(\lambda g(x),g(x))$ . I guess (c) will be false. Not sure though.

REPLY [4 votes]: I am sure first two thirds of this question has been discussed hee already..
Coming to third part I would go for a very basic example of linear map :
$f,g:\mathbb{R}\rightarrow \mathbb{R}$ defined as $f(x)=x;g(x)=2x$
This should conclude $c$ is True/False??