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Solved: A12 An A2n A21 A22 Problem 2. Given An N X N Matrix A = We Define The Trace Of A, Denoted : ??n An2 Anl Tr(A), By N Tr(A) = Aii A11 +:::+ann- I=1 (a) Prove That, For Every N X M Matrix A And F

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a12 an a2n a21 a22 Problem 2. Given an n x n matrix A = we define the trace of A, denoted : ??n an2 anl tr(A), by n tr(A) = a



a12 an a2n a21 a22 Problem 2. Given an n x n matrix A = we define the trace of A, denoted : ??n an2 anl tr(A), by n tr(A) = aii a11 +:::+ann- i=1 (a) Prove that, for every n x m matrix A and for every m x n matrix B, it is the case that tr(AB) 3D tr(??). tr(A subspace V C R". Prove that norm (b) Let (c) Let P be the matrix of an orthogonal projection from R" to a tr(P) Prove an n x m matrix mns are = m. dim(V) =

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Let A = [a_11 a_12 . . . a_1m a_n = 1 . . . a_nm]_n times m B = [b_11 b_12 . . . b_1n b_m1 . . . b_mn]_m times n Let AB = [(AB)_11 (AB)_12 . . . . (AB)_1 times n (AB)_n1 . . . (AB)_nn]_n times

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