1. ## Linear functional problem

I have been strugling with this problem for a while now, would appreciate any direction.

If $\displaystyle l\neq 0$ is a linear functional on an n-dimensional vector space V.
Show that there exists a basis of V $\displaystyle B=\{v_{1},...v_{n}\}$, such that for every $\displaystyle v\inV$, if $\displaystyle v=\alpha_{1}v_{1}+...+\alpha_{n}v_{n}$, then $\displaystyle l(v)=\alpha_{1}$.

My only idea so far is that if I can show that there exists a basis of V $\displaystyle B=\{v_{1},...v_{n}\}$ such that $\displaystyle l(v_{1})=1$ and $\displaystyle l(v_{i})=0$ for $\displaystyle i=2,...,n$ then I will have what I need. I was thinking of chossing a $\displaystyle v$ such that $\displaystyle l(v)=1$ and then completing that to an orthonormal basis, but I can seem to get anywhere with this.

2. Originally Posted by skyking
I have been strugling with this problem for a while now, would appreciate any direction.

If $\displaystyle l\neq 0$ is a linear functional on an n-dimensional vector space V.
Show that there exists a basis of V $\displaystyle B=\{v_{1},...v_{n}\}$, such that for every $\displaystyle v\inV$, if $\displaystyle v=\alpha_{1}v_{1}+...+\alpha_{n}v_{n}$, then $\displaystyle l(v)=\alpha_{1}$.

My only idea so far is that if I can show that there exists a basis of V $\displaystyle B=\{v_{1},...v_{n}\}$ such that $\displaystyle l(v_{1})=1$ and $\displaystyle l(v_{i})=0$ for $\displaystyle i=2,...,n$ then I will have what I need. I was thinking of chossing a $\displaystyle v$ such that $\displaystyle l(v)=1$ and then completing that to an orthonormal basis, but I can seem to get anywhere with this.

Look at $\displaystyle \ker l$ . This is a subspace of dimension n -1, so choose from here a base $\displaystyle \{v_2,\ldots,v_n\}$ complete it to a base $\displaystyle \{v_1,\ldots,v_n\}$

of the whole space, and since then $\displaystyle l(v_1)=k\in\mathbb{F}\,,\,k\neq 0$, instead $\displaystyle v_1$ take $\displaystyle \frac{1}{k}v_1$

Tonio