As we saw in Chapter 3, there are many vector spaces. Naturally, one can ask whether or not two vector spaces are the same. To say two vector spaces are the same or not, one has to compare them first as sets, and then see whether or not their arithmetical rules are preserved. A usual way of comparing two sets is defining a function between them. Recall that a function from a set X into another set Y is a rule which assigns a unique element y in Y to each element x in X. Such a function is denoted as f : X → Y and sometimes referred to as a transformation or a mapping. We say that f transforms (or maps) X into Y. When given sets are vector spaces, one can compare their arithmetical rules also by a transformation f if f preserves the arithmetical rules, that is, f (x + y) = f (x) + f (y) and f (k x) = kf (x) for any vectors x, y and any scalar k. In this chapter, we discuss this kind of transformations between vector spaces via the linear equation A x = b.
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