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&#039;=\{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.