Let me re-enforce this by looking at a similar problem.
Suppose f is a linear transformation from to defined by .
To prove it is a linear mapping you must show that f(u+ v)= f(u)+ f(v) and f(av)= af(v) for any vectors u and v and any number a. You need to show that
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Also, .
To write it as a matrix, in a given basis, apply the transformation to the basis, the right the result of each as a linear combination of the basis. Each basis vector gives a column of the matrix. The standard matrix is and .
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So the matrix representation of A in the standard basis is