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I am trying to understand the matrix representation of a linear transformation.

So here is my thought process.

Let B = (b1, b2, ..., bn) be a basis for V, and let Y = (y1, y2, ..., ym) be a basis for W.

T: V --> W

Pick and v in V and express as a linear combo of the basis vectors:

v = sum( ai bi, 1, n)

T(v) = sum( ai T(bi), 1, n)

i.e., the transformed vector T(v) is determined by a linear combination of the transformed basis vectors.

Now coordanitize everything relative to Y, which we can always do since it is an isomorphism.

[T(v)]_Y = sum( ai [T(bi)]_Y, 1, n)

Then we can write this linear combination as a matrix multiplication, i.e., the vectors [T(bi)]_Y give the column vectors of the matrix representation.

Anyway, it took me awhile to get this and I still doubt myself. Is my reasoning correct?