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Are we able to represent linear matrix transformations with complex numbers?

Absolutely. Consider a point (a,b). This may be represented by the complex number a+bi and also by the column vector (a;b), where the semicolon means 'new line'.

To translate the point by +(c,d), in complex numbers, this is done by adding c+di to a+bi.  In 2D space, this is done by adding (c;d) to (a;b).

To scale the point by a factor of r, in complex numbers, this is done by multiplying by r.  In 2D space, we do the very same thing.

To rotate the point about (0,0) by angle t in the counterclockwise direction, in complex numbers, we do this by multiplying by e^it. In 2D space, we multiply on the left hand side by the matrix ((cost,-sint);(sint,cost)).

To conclude, if we were to translate a point (a,b) by +(c,d), scale it by factor r and rotate it about the origin by angle t in the counterclockwise direction, then the following are two representations of it:

(a+bi)(re^it)+(c+di)

r((cost,-sint);(sint,cost))(a;b)+(c;d)

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