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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.

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)

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up vote 2 down vote accepted

This does not follow (at least not directly) from Kronecker-Capelli, since that tells you that with a full rank $(n-k)\times n$ matrix, the solutions of the corresponding homogeneous system of equations will define some subspace of dimension $k$, not necessarily the one $V$ you are after. You somehow have to find your matrix, given $V$. There are many ways to proceed, and you can use Kronecker-Capelli if you like. Think of the space of all linear forms on $K^n$ (in other words $1\times n$ matrices) that vanish on $V$ (giving equations that could be used to define $V$).

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