I am getting ready for my friday exam, and I am not able to solve this formally. Let me present you my attempt and, please, check where it lacks in logic.
Question:
Prove, that every $k$ dimensional subspace $V\subset \mathbb{K}^n$ can be described with $n-k$ linear equations.
So: Using Kronecker-Capelli, we know that$\dim V=n -r(K).$ If subspace $r $ dimensional can be described in $\mathbb{K}^r $ it must have $r$ unique linear equations. Is it not straight forward from the theorem? If it is, is the question about proving Kronecker-Capelli? If not, emm... what is it about? Thank you in advance for help.
Does anyone see standard (way before homogeneous systems) proof? I put emphasis on the fact, that it is not my homework, just want to learn standard ways of proving (and get ready for an exam)