TITLE: Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.9, Problem 12
QUESTION [6 upvotes]: If $f_1, \ldots, f_p$ are linear functionals on an $n$-dimensional vector space $X$, where $p<n$, then how to show that there is a vector $x \ne 0$ in $X$ such that $f_1(x) = 0, \ldots, f_p(x)=0$? What consequences does this result have with respect to linear equations? 
My feeling is that one of the consequences of this result with respect to linear equations is the following fact: 
Any system of $p$ homogeneous linear equations in $n$ unknowns, where $p < n$, must have a non-trivial solution. 
Am I right? 
And if so, then what other consequences does our result entail?

REPLY [5 votes]: Suppose that there does not exist such $x$. Then the mapping $\phi : X\to \mathbb R^p$ defined by $$\phi (x)=(f_1(x),\cdots,f_p (x))$$ is injective. Therefore, $dim X\leq dim \mathbb R^p$  or $n\leq p$. A contradiction.