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

In summary, the conversation discusses the proof that if m < n and there are m linear functionals on an n-dimensional vector space V, there exists a non-zero vector x in V such that [x,y_j]=0 for j=1,\cdots,m. This result implies that the solutions of linear equations consist of a homogeneous solution and a particular solution.
  • #1
Dafe
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Homework Statement


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


Homework Equations



N/A

The Attempt at a Solution



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

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

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

Any input is very welcome, Thanks!
 
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  • #2
Gentle bump.
 

What is the meaning of the statement "If m

In this statement, m and n refer to the number of elements in a set. If m

What is the significance of proving that y_1, ..., y_m are linear functionals?

Proving that y_1, ..., y_m are linear functionals means that for any given set of inputs, these functionals will produce a linear output. This is important in mathematical analysis and applications in fields such as physics and engineering.

How can one prove that y_1, ..., y_m are linear functionals?

This can be done by applying the definition of a linear functional, which states that for any two vectors x and y, and any scalar c, the following must hold: y_i(cx + y) = cy_i(x) + y_i(y). If this condition is satisfied for all y_i, then the functionals are linear.

What are the implications of m

If m

Can y_1, ..., y_m still be linear functionals if m>n?

Yes, it is possible for y_1, ..., y_m to be linear functionals even if m>n. However, this may result in some functionals being dependent on others, which can complicate the analysis and interpretation of the system. In most cases, it is more desirable for m

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