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Linear Algebra : Linear Transformations, Vector Space and Basis

Let C^2x2 be the complex vector space of 2 x 2 matrices with complex entries. Let B= [1 -1]
[-4 4]

and let T be the linear operator defined on C^2x2(T: C^2x2 --> C^2x2) by T(A)=BA. What is the rank of T? Can you describe T?

For this question you will need to get a basis of C^2x2 and use it to write a matrix representative of T. Recall that a linear operator is a linear transformation from some vector space V into V.
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Linear Transformations, Vector Space and Basis are investigated.

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