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  3. 2 let m3x3 be the vector space of all 3...

Question: 2 let m3x3 be the vector space of all 3...

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2. Let M3x3 be the vector space of all 3 x 3 matrices with real entries. Let E = {A E M3x3 I all row and column sums of A are equal). For example L E where 1-3 L 2 0 0 1 -25 because all rows and columns of L sum to 2. A basis for E is where 1 -1 0 0 1-1 -1 1 0 0-11 and B5 -I, the identity matrix. Do the following problems (which are similar to the Lab D problems)(a) Find [L]s, the coordinate vector of L relative to the basis B. Hint: to express L in terms of the basis vectors, start by writing where cs is a scalar and Z E W. Here W is the subspace spanned by { Bi, B2, Bs. B4} (so all rows and columns of Z sum to 0.) b) The trace on M3xs restricted to the subspace E is still a linear transfor- mation. Represent as a matrix [tr]ss. This is the matrix of the trace (as a linear transforma tion from E to R) relative to the bases B and S. Here Ss the standard basis for R. i. What is the size of [tr]ss? ii. Compute [trlss- iii. Check that [trlss L]s [tr(L)s

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