Let V be a vector space on a field $K$ and be $T: V → K$ a non-zero linear application. Show that if W a subspace of V of dimension $n - 1$ then there is a linear application $T: V → K$ such that $W = ker T$.
My attemp is: take a T(v)= $a_{1}$$x_{1}$+$a_{2}$$x_{2}$+...+$a_{n}$$x_{n}$ Note that T is epimorphism, by the dimension theorem, then $W = ket(T)$. I need a help. How would you do it?. I'm very confused.
Fix a basis for $W$, say $x_1,...,x_{n-1}$ and extend this to a basis of $V$ by adding in $x_n$. Then define $T(v) = T(\sum_i a_i x_i) = a_n $ where $a_n$ is the coefficient of $x_n$. Your map does not necessarily work, because while $T$ has a kernel of dimension $n-1$ there is no guarantee that this kernel is $W$. You are given $W$ and need to construct a $T$ that vanishes on $W$.