Definition of linear dependence for a finite set of functions.

Question

I was asked to define linear dependence for a finite set of functions. Please note that this is differential equations class, not Linear Algebra.

Can someone please verify whether these definitions are correct? Thanks!

Here it is:

I first defined the term 'linear combination' of functions:

Linear Combination: The function $c_1y_1 + c_2y_2 + ... + c_ny_n$ with arbitrary numerical values $c_1, c_2, ..., c_n$ is called a linear combination of the functions $y_1,y_2,...,y_n$.

Linear Dependence: A set of functions $\{y_1, ... , y_n\}, y_i : I \subset \mathbb R$, is linearly dependent iff you can write one function as a linear combination of the others. More precisely, iff $c_1y_1 + ... + c_ny_n = 0$ where not all $c_1, c_2, ... , c_n$ are zero.

Answer 1

I see no problem about this definition, they should be right.

One comment is that (and is already in your definition) is that although we have large number of functions (uncountable, good that you said the index set is a subset of $\mathbb R$), the definition for linear combination and dependence is limit to finite sum.

The good thing about finite sum is that it is clearly defined, and does not require any more topological structure to support concepts like "dense" or "limit".

Answer 2

Your definitions are fine. There is one small objection that may be observed in your definition of linear dependence, depending on how pedantic your mentor may be (I know mine was). You may want to say

A set of functions $\{y_1,\ldots,y_n\}$ with domain $I$ is said to be linearly dependent on $I$ if there exist scalars $c_1,\ldots,c_n$ not all zero, such that $$c_1y_1(t)+\cdots+c_ny_n(t)=0$$ for all $t\in I$.

That is, you may want to mention that the linear combination is zero for all inputs.