Let \(X\) be a set together with operations of addition, \(+ \colon X \times X \to X\), and multiplication, \(\cdot \colon {\mathbb{R}}\times X \to X\), (we write \(ax\) instead of \(a \cdot x\)). \(X...Let \(X\) be a set together with operations of addition, \(+ \colon X \times X \to X\), and multiplication, \(\cdot \colon {\mathbb{R}}\times X \to X\), (we write \(ax\) instead of \(a \cdot x\)). \(X\) is called a vector space (or a real vector space) if the following conditions are satisfied: (Addition is associative) If \(u, v, w \in X\), then \(u+(v+w) = (u+v)+w\). (Addition is commutative) If \(u, v \in X\), then \(u+v = v+u\). (Additive identity) There is a \(0 \in X\) such that \(v+0=v\)…