Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here's my definition of the determinant:

If $t:U \to U$ is a linear operator and $\dim(U)=n$ then $\det(t)$ is the unique number satisfying $$tu_1 \wedge \cdots \wedge tu_n = \det(t) u_1 \wedge \cdots \wedge u_n$$ for all $u_1,\dots,u_n \in U$.

Define the transpose $\tau^T : V^* \to V^*$ of $\tau$ by $(\tau^Tf)(v)=f \tau v$. Is there a way of proving that $\det(\tau^T)=\det(\tau)$ without choosing a basis? It's clear that

$$\det(\tau) v_1 \wedge\cdots\wedge v_n = \tau v_1 \wedge\cdots\wedge \tau v_n$$ and $$\det(\tau^T) f_1 \wedge\cdots\wedge f_n = \tau^T f_1 \wedge\cdots\wedge \tau^T f_n$$ for all $v_1,\dots,v_n \in V$ and $f_1,\dots,f_n \in V^*$, but how do I "join" them together?

  • $\begingroup$ I don't understand $\wedge^k V$ is now one dimensional... $\endgroup$ – user38268 Nov 5 '12 at 12:40
  • $\begingroup$ @BenjaLim: What do you mean? $\endgroup$ – wj32 Nov 5 '12 at 12:43

Define the "alternating" multilinear map \begin{align} (V^*)^n \times V^n &\to F \\ (f_1,\dots,f_n,v_1,\dots,v_n) &\mapsto \det(f_i(v_j)) \end{align} where $\det(f_i(v_j))$ refers to the determinant of the matrix with its $(i,j)$ entry equal to $f_i(v_j)$. This gives us (details omitted) a nondegenerate pairing $\langle\cdot,\cdot\rangle:\bigwedge^n V^* \times \bigwedge^n V \to F$ with $$(f_1 \wedge\cdots\wedge f_n,v_1 \wedge\cdots\wedge v_n) \mapsto \det(f_i(v_j)).$$ So for all $f_1,\dots,f_n \in V^*$ and $v_1,\dots,v_n \in V$, \begin{align} \det(\tau)\langle f_1 \wedge\cdots\wedge f_n, v_1 \wedge\cdots\wedge v_n \rangle &= \langle f_1 \wedge\cdots\wedge f_n, \det(\tau)v_1 \wedge\cdots\wedge v_n \rangle \\ &= \langle f_1 \wedge\cdots\wedge f_n, \tau v_1 \wedge\cdots\wedge \tau v_n \rangle \\ &= \det(f_i\tau v_j) \\ &= \det((\tau^T f_i)v_j) \\ &= \langle \tau^T f_1 \wedge\cdots\wedge \tau^T f_n, v_1 \wedge\cdots\wedge v_n \rangle \\ &= \det(\tau^T)\langle f_1 \wedge\cdots\wedge f_n, v_1 \wedge\cdots\wedge v_n \rangle. \end{align}

Therefore $\det(\tau)=\det(\tau^T)$.

  • $\begingroup$ @wj32.Why is there exist the pairing $<.,.>$ ? How can I prove the existence via alternating maps ($\textbf{without}$ using notion of tensor product of vector spaces) as I gave it in math.stackexchange.com/questions/1364235/… ? $\endgroup$ – bigli Jul 20 '15 at 6:05

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