Theorem A(x + y) = Ax + Ay, A(cx) = cAx.
Proof By an earlier exercise, we have
x1a1 + x2a2 + ... + xnan = Ax
where A = [a1 a2 ... an] is a matrix with the vectors a1, a2, ..., an as columns, and x1, x2, ..., xn are the coordinates of x. Thus
Ax + Ay
+ ... + xnan
+ ... + ynan
= (x1a1 + y1a1) + (x2a2 + y2a2) + ... + (xnan + ynan)
= (x1 + y1)a1 + (x2 + y2)a2 + ... + (xn + yn)an
Since x1 + y1, x2 + y2, ..., xn + yn are the coordinates of x + y, we see that the summation is A(x + y).
The proof for A(cx) = cAx is similar.