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Let ##T: \mathbb{R^2} \to \mathbb{R^2}##. Suppose I have standard basis ##B = \{u_1, u_2\}## and another basis ##B^{\prime} = \{v_1, v_2\}## The linear transformation is described say as such ##T(v_1) = v_1 + v_2, T(v_2) = v_1##

If I want to write the matrix representing ##T## with respect to basis ##B^{\prime}## then I'll just find ##[T]_{B'}##. I can also find ##[T]_{B}## rather straightforward using similarity transformation if I know the transition matrix between those two bases.

But suddenly I encounter this notation ##[T]_{B,B'}##. I don't know exactly what this notation represents. Do you guys know what this notation mean? What other matrix should I provide in this case? Normally I use that comma subscript to denote transition matrix between bases, but never for linear transformation matrix.

Thank You