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Question: let m33 be the vector space of all 3 3...

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Let M33 be the vector space of all 3 3 matrices with real entries. Let E = fA 2 M33 j all row and column sums of A are equalg. For example L 2 E where

Let M3x3 be the vector space of all 3 x 3 matrices with real entries. Let E = {A E M3x3 l all row and column sums of A are equal. For example LEE where L 20 0 1 -2 5 because all rows and columns of L sum to 2. A basis for E is where 3 ー1 0 -1 1 and B5 - I, the identity matrix. Do the following problems (which are similar to the Lab D problems):

(a) Find [L]B, the coordinate vector of L relative to the basis B. Hint: to express L in terms of the basis vectors, start by writing where C5 is a scalar and Z є W. Here W is the subspace spanned by (Bl. B2. B3. Bi} (so all rows and columns of Z sum to 0.) (b) The trace on M3x3 restricted to the subspace E is still a linear transfor- ation. Represent as a matrix trs. This is the matrix of the trace (as a linear transforma- tion from E to R) relative to the bases B and S. Here S-(l] is the standard basis forIR i. What is the size of trsB? ii. Compute [trlsa iii. Check that trlsBlLB=[T(L)|s.

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