Identify all linear transformations from C2 to C3

[jinksys]

Homework Statement

Homework Equations

The Attempt at a Solution

In the previous problem I was asked to identify if a polynomial, such as f(x)=2x was a linear transformation. In that case I checked to see if f(ax + by) = f(ax) + f(by). I figure I would be doing something similar in this case.

I would check to see if T(ax + by) = T(ax) + T(by).

So for problem (A) would I do something like:

and since T(ax + by) ≠ T(ax) + T(by), T1 is not a linear transformation?

 

[Bacle]

You are correct, T1 is not a linear transformation, and your argument is correct/valid.

 

[jinksys]

Thanks!

 

[jinksys]

I have a question about part (D), which is

$$<br /> Tx =<br /> \left[ {\begin{array}{cc}<br /> 1 & 0 \\<br /> 2-i & 3i \\<br /> i-i & 4\\<br /> \end{array} } \right]x$$

I assume I use the same procedure? Is anything special I need to do in regards to the complex variables?

I find that it IS a linear transformation, is that correct? Also, isn't i-i just 0?