Theorem

A homogenous system of linear equations with more variables than equations has nonzero solutions.

Proof

We use the following notation. Fix positive integers m and n, and let ${latex.inlineA_{j,k} \in F} for ${latex.inlinej = 1, ..., m} and ${latex.inlinek = 1, ..., n}. Consider the following homogenous system of linear equations:

${latex.inline\sum_{k=1}^{n}A_{1,k}x_{k} = 0} . . . ${latex.inline\sum_{k=1}^{n}A_{m,k}x_{k} = 0}

m describes how many linear equations there are, n describes how many variables(in this case x’s) each linear equation works with.

We can define a linear map ${latex.inlineT: F^{n} \rightarrow F^{m}} by defining ${latex.inlineT(x_{1}, ..., x_{n}) = (\sum_{k=1}^{n}A_{1,k}x_{k}, ..., \sum_{k=1}^{n}A_{m,k}x_{k})}. In English, this means that we map each vector ${latex.inline(x_{1}, ..., x_{n})} to a vector which contains the value of each equation as its elements. For example, the first element is the result of the components of the x vector plugged into the first equation.

With this framing, the equation ${latex.inlineT(x_{1}, ..., x_{n}) = 0} is the same as expressing the homogenous system of linear equations. We are interested in whether null T is strictly bigger than {0}, which would mean that there is a non zero solution to the system of linear equations. This is equivalent to T being not injective.

Well, if the system of linear equations has more variables than equations, we know that n is greater than m, and T is a linear map from a larger dimension to a smaller one. Per 3.22, this is not injective, and thus there is at least one non zero solution.