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Assume that $E$ is a vector space over $\mathbb K$ (which is $\mathbb R$ or $\mathbb C$). If $(f_{1},…,f_{k}) \in L(E,\mathbb K)=E^*$ (we consider $k$ linear maps from E to $\mathbb K$). I am wondering how to prove that $$ \dim \cap_{i=1}^{k} Ker \ f_{i}=n-k \Leftrightarrow \{f_1,…,f_{k}\} \ \rm{are \ linearly \ independent\ in} \ E^* $$ It is mentioned in a book of linear algebra without proof and this lemma seems to be useful, for instance to prove that a family of linear forms is linearly independent. The proof looks very difficult for me. I am undergraduate student. Thanks for any help.

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1 Answer 1

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Here's one way to think about it. Write elements in $E$ as column vectors, and elements in $E^*$ as row vectors. Then given $f_1,\dots,f_k \in E^*$ we can form the matrix $M$ whose rows are the $f_i$s: $$ M = \begin{pmatrix} \leftarrow f_1 \rightarrow \\ \leftarrow f_2 \rightarrow \\ \vdots \\ \leftarrow f_k \rightarrow \end{pmatrix}. $$

Then observe that

$$\text{ker} M = \cap_{i=1}^k \text{ker} f_i.$$

What is the rank of $M$ when:

  • $\text{dim} \cap_{i=1}^k \text{ker} f_i = n-k$?
  • $f_1,\dots,f_k$ are linearly independent?
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