Linear application in canonical basis

Question

I have the function $f:\mathbb C^3 \to \mathbb C^3$, $f(x_1,x_2,x_3) = (4x_1 + 2x_2, x_1+4x_2+x_3, x_1+x_2+4x_3)^t$. $t$ stands for transposed

How can I determine if it is an linear application and it's matrix in a canonical basis?

Answer 1

Just use the definition of linearity to check if this holds for the given function.

To find the corresponding matrix using the canonical basis take you should calculate $f(1,0,0), f(0,1,0)$ and $f(0,0,1)$ (that is the images of the canocical basis vectors) which gives you the columns of the matrix. The article on wikipedia might also be of interest to you.

Answer 2

You can think that your function $f:{\Bbb C}^3\longrightarrow{\Bbb C}^3$ can be expressed as $$x=\left(\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right) \longmapsto f(x)= \left(\begin{array}{ccc} 4&1&1\\ 2&4&1\\ 0&1&4 \end{array}\right) \left(\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right),$$ so, by attending the above Hirshy and Keenan Kidwell's recommendation's you can settle your questions.