How to prove, that any vector subspace of dimension $k$ of vector space of dimension $n$ is intersection of kernels of some $n-k$ linear functions.
Proving, that any vector subspace of dimension $k$ of vector space of dimension $n$ is intersection of kernels of some $n-k$ linear functions
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linear-algebra
vector-spaces
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Let $\{u_1, \dots, u_k\}$ be a basis for the subspace and $\{u_1, \dots, u_k, v_{k+1}, \dots, v_n\}$ be a basis for the whole space. For $i =k+1, \dots, n$, define the linear functional on the basis as follows: $f_{i}(v_{i}) = 1$ and $f_{i}$ maps other elements of the basis to 0. Then the subspace is the intersection of kernel of these functionals.