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# hw2 - HOMEWORK 2 1(a Let Snn(C denote the set of all n n...

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HOMEWORK 2 1. (a) Let S n × n ( C ) denote the set of all n × n symmetric matrices with complex entries. Prove that this set is a subspace of M n × n ( C ). Find a basis for this subspace, and give a formula for its dimension (in terms of n ). (b) Recall that for a square matrix A , tr( A ) denotes the sum of the diagonal entries of A . Prove that { A M 2 × 2 ( C ) : tr( A ) = 0 } is a subspace of M 2 × 2 ( C ) Find a basis for this subspace and compute its dimension. (c) Let W n = { p ( x ) P n ( C ) : p ( - 1) = p (1) } . Show that this is a subspace of P n ( C ). Find a basis for it, and give a formula for its dimension (in terms of n ). 2. (a) Let S be a collection of sets (that is, every element of S is itself a set). Then we can define the intersection of the sets in S (even if S is an infinite collection of sets), as follows: \ X ∈S X := { x : X ∈ S ( x X ) } Show that if V is a vector space and W is a subspace of V for each W in some collection S , then T W ∈S W is also a subspace of V .

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