If m<n prove that y_1, ,y_m are linear functionals

[Dafe]

Homework Statement

Prove that if m<n, and if $$y_1,\cdots,y_m$$ are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that $$[x,y_j]=0$$ for $$j=1,\cdots,m$$. What does this result say about the solutions of linear equations?

Homework Equations

N/A

The Attempt at a Solution

Let $$X=\{x_1,\cdots,x_n\}$$ be a basis for V.
By a theorem in the book I know that there exists a uniquely determinet basis X' in V',
$$X'=\{y_1,\cdots,y_n\}$$ with the property that $$[x_i,y_j]=\delta_{ij}$$.

Every $$x\in V$$ can be written as $$x=\xi_1 x_1+\cdots+\xi_n x_n$$.
If we let $$j=1,\cdots,m$$ we get,
$$[x,y_j]=\xi_1[x_1,y_j]+\cdots+\xi_m[x_m,y_j]$$ and since $$[x_i,y_j]=\delta_{ij}$$, there exists some x such that $$[x,y_j]=0$$.

The result says that the solution of linear equations consists of a homogeneous solution and a particular solution.

Any input is very welcome, Thanks!

 

[Dafe]

Gentle bump.