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Linear Algebra: Adjoint
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[QUOTE="Fredrik, post: 4707255, member: 14944"] Yes, it looks fine. You didn't post the formula ##(M_T)_{ij}=(Te_j)_i##, but since your ##M_T## is consistent with it, I assume that you were using it. One of your options to calculating ##M_{T^*}## as the conjugate transpose of ##M_T##, is to start with ##(M_{T^*})_{ij}=(T^*e_j)_i##, and then rewrite the right-hand side as an inner product and use the definition of the adjoint operation. Since the problem is asking you to find ##T^*x## for an arbitrary ##x\in\mathbb R^2##, you don't have to explicitly write down a matrix. You can just start like this: $$T^*x=T^*\left(\sum_j x_j e_j\right)=\sum_j x_j T^*e_j=\sum_j x_j\sum_i(T^*e_j)_i e_i,$$ were ##(T^*e_j)_i## is defined as the ith component of the vector ##T^*e_j## with respect to the ordered basis ##(e_1,e_2)##. This is of course the ij-component of the matrix of ##T^*##, but you don't have to know that to continue this calculation. [/QUOTE]
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