Homework #8 for Math H110, fall '02
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I'm going to use @ to denote tensor product.
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1. If pi is a permutation of the numbers 1 through n, and VxVx...xV is the
set of n-tuples (v_1,v_2,...,v_n), define pi.(v_1,v_2,...,v_n) to be
(v_{pi(1)}, v_{pi(2)}, ..., v_{pi(n)}), rearranging the n-tuple.
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Since VxVx...xV is the basis for V*V*....*V, we can linearly extend this
definition to V*V*...*V. Show that each pi preserves the subspace I of
V*V*...*V. Therefore, pi gives a well-defined map from V @ V @...V.
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2. If V is n-dimensional, define an isomorphism of V @ V with nxn matrices,
in such a way that "apply the permutation 1<->2" (from question 1) turns
into "transpose".

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3. Define Sym^2 V = { t in V@V: (1<->2).t = t }. What's its dimension?

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4. Define the map Alternate : V@V@...@V to itself by 
<center> t |-> sum_{pi a permutation of 1..n} (-1)^pi  pi.t </center>
(here (-1)^pi means like the +/-1 we got in the determinant formula).
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Define Alt^n V to be the image of this map, inside V@V@...@V (n factors).
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Given a basis v_1...v_d of V, define an "ordered pure tensor" as one of
the form v_(i_1) @ v_(i_2) @ .. @ v_(i_d), where i_1 < i_2 < ... < i_d.
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Let R be the subspace of V@V@...@V spanned by the ordered pure tensors.
Show that Alternate : R -> Alt^n V is an isomorphism.
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5. What's the dimension of Alt^n V?

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6. If T : V->W, find a natural map from Alt^n V -> Alt^n W.