Let $T_1$ and $T_2$ be two linear transformations from $R^n$ to $R^1$. If $ker(T_1)=ker(T_2)$, show that there is a non-zero constant $a$ such that $T_1(v)=aT_2(v)$ for all $v\in R^n$
Analysis: I need to achieve $T_1(v)=aT_2(v)$, that is $Av=aBv$
Since $T_1$ and $T_2$ are mappings from $R^n$ to $R^1$, they must both be $1\ X\ n$ row matrices. (Not sure how this will help.)
Let the standard matrix for $T_1$ and $T_2$ be A and B respectively. Therefore, $ker(T_1)$ is the solution space of $Ax=0$ and similarily, $ker(T_2)$ is the solution space of $Bx=0$
I am not making any progress. Can anyone guide me in the right direction? Thanks Stack!@ Owe you one again and again.