3x3-matrices/Determinant 0/Linear subspace/Dimension 7/Exercise

Let ${}K$ be a field, and let

${}T={\left\{M\in \operatorname {Mat} _{3\times 3}(K)\mid \det M=0\right\}}\subset \operatorname {Mat} _{3\times 3}(K)=K^{9}\,$

be the set of all ${}3\times 3$-matrices with determinant ${}0$. Does ${}T$ contain a linear subspace of dimension ${}7$?

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