Linear operators are mathematical entities that act on vectors within a vector space, producing another vector in the same or another vector space while satisfying two key properties: additivity and homogeneity. This means that when a linear operator is applied to the sum of two vectors, it equals the sum of the operator applied to each vector separately, and when it is applied to a scalar multiple of a vector, the output is the scalar multiplied by the result of the operator applied to that vector. These operators play a crucial role in understanding transformations in quantum mechanics and can be represented by matrices in finite-dimensional spaces.
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