Let $V$ be a vector space of dimension $n$ and ${v_1,v_2,...,v_n}$ be a basis of $V$. Let $\sigma \in S_n$ and $T:V \rightarrow V$ be linear transformation defined by $T(v_i)=v_\sigma(i)$. Then what can you say about $T$ ?
2026-03-27 01:42:25.1774575745
Linear Transformation on Permutation Set
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Null space of $T$ consists of the $0$ vector! Also the matrix of the linear transformation with respect to this basis consists of only $0$'s and $1$'s and has $n$ linearly independent rows and columns.