The complex numbers C can be identified with the vector space R^2 via the identification c + di ---> [c, d].

Let w = a + bi define the map T:C---%26gt;C as T(z) = wz.





a) Use the identification of C with R^2 that is mentioned above and show that T can be viewed as a linear transformation from R^2 to R^2.





b) find the matrix that represents T with respect to the standard basis of R^2.

The complex numbers C can be identified with the vector space R^2 via the identification c + di ---%26gt; [c, d].
When you multiply out (a + bi) * (x + yi) and rephrase it in terms of ordered pairs, you get





T (x,y) = (ax - by, bx + ay).





So the matrix of T has a -b for its first row and b a for its second row.