Terminology with linear transformation

Question

I am working on a problem that asks me to "Write C for the matrix whose ij entry is $(1/2)^{ij}$" given that $M$ is the vector space of all $n x n$ matrices and $l$ is a linear transformation on $M$. What does it mean to write C?

Answer 1

This is just defining a matrix, the phrase "Write $C$ for BLAH" just means that BLAH is some statement that defines a mathematical object and $C$ is the letter we will use to refer to that object. For example "write C for the square of $3$" just means "$C = 3^2$".

In your case BLAH is "the matrix whose $ij$ entry is $(\frac12)^{ij}$. This is defining a matrix by telling you what the entries of that matrix are. For example the entry in the second row and third column (the $(2, 3)$ entry) is $(\frac12)^{2\cdot3} = (\frac12)^6 = \frac{1}{64}$.

The only information you're not given is the size of the matrix $C$. Presumably this should be clear from context.