Linear map from n-dim space to p-dim space

[hooyeh]

I've been thinking about the following question:
if [tex]x\in R^n[/tex] and [tex]y=Cx\in R^p[/tex] where matrix [tex]C[/tex] describes the linear map from n dimensional reals to p dimensional reals. If we only have access to [tex]y[/tex] and want to recover the information about [tex]x[/tex], which components in [tex]x[/tex] are needed? I kind of figured out that the number of components that we need is [tex]n-rank(C)[/tex].

For example, [tex]x=[x_1; x_2; x_3][/tex] and [tex]y=[x_1; x_1+x_2+x_3][/tex]. Obviously, either [tex]x_2[/tex] or [tex]x_3[/tex] could help us recover the information of [tex]x[/tex]. Another trivial example, [tex]x=[x_1; x_2; x_3][/tex] and [tex]y=[x_1; x_1][/tex], then we need both [tex]x_2[/tex] and [tex]x_3[/tex].

But how to state this observation rigorously and neatly in the mathematical language for any two such vectors [tex]x\in R^n[/tex] and [tex]y=Cx\in R^p[/tex]?

Any helpful suggestions are appreciated!

 

[lippyka]

A formal way to state this observation is that the number of components of x needed to recover the information of x is equal to n-rank(C). This can be proven using linear algebra. The proof works by showing that the system of equations of the form y=Cx has a unique solution only when the rank of C is equal to n, which means that all components of x are necessary for a unique solution. When the rank of C is less than n, then some components of x are not necessary for a unique solution and can be omitted.