Let V , W be finite dimensional vector spaces over R. Let A : V->W be a linear map. Choose bases of V and W and the corresponding bases of $\Lambda^k$(V ) and of $\Lambda^k$(W). How to show that the entries of the matrix representing $\Lambda^k$(A) are polynomial in the entries of the matrix representing A.
Matrix representing $\Lambda^k$(A)
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tensors
1 Answers
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Apply $\Lambda^k(A)$ to any basis element $v_{i_1} \wedge \ldots \wedge v_{i_k}$, and you get $Av_{i_1} \wedge \ldots \wedge Av_{i_k}$. Write each $Av_{i_j}$ in terms of the basis for $W$. Expand, getting the coefficient of each term being a product of elements of $A$. Collecting terms to get an expression in the basis for $\Lambda^k(W)$. You get coefficients being sums of product of elements of $A$. Ie, polynomials in elements of $A$.