A homogenous system of linear equations with more variables than equations has nonzero solutions.
We use the following notation. Fix positive integers m and n, and let \({latex.inline[A_{j,k} \in F](A_{j,k} \in F)} for \){latex.inlinej = 1, ..., m} and ${latex.inlinek = 1, ..., n}. Consider the following homogenous system of linear equations:
\({latex.inline[\sum_{k=1}^{n}A_{1,k}x_{k} = 0](\sum_{k=1}^{n}A_{1,k}x_{k} = 0)} . . . \){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 \({latex.inline[T: F^{n} \rightarrow F^{m}](T: F^{n} \rightarrow F^{m})} by defining \){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 1756253983 - Axler 3.22 A linear map to a lower dimensional space is not injective.|3.22, this is not injective, and thus there is at least one non zero solution.