Question

1. Suppose v,w element of V. Explain why there exists a unique x element of V such that v +3x = w.

Answer 1

Suppose $v,w$ element of $V$. Let us show that there exists a unique $x$ element of $V$ such that $v +3x = w$ . Let $x_1$ and $x_2$ are the solutions of the equation $v +3x = w$. Then $v +3x_1 = w$ and $v +3x_2 = w$. It follows that $v +3x_1 =v +3x_2.$ Since $(V,+)$ is a group, using left cancellation law we conclude that $3x_1 =3x_2$. After multiplying by $\frac{1}{3}$ from left we have that $\frac{1}{3}(3x_1) =\frac{1}{3}(3x_2).$

Then using compatibility of scalar multiplication with field multiplication we conclude that $(\frac{1}{3}3)x_1 =(\frac{1}{3}3)x_2,$ and hence $1\cdot x_1=1\cdot x_2.$ Using property of identity element of scalar multiplication we conclude that $x_1=x_2.$ Consequently, there exists a unique $x$ element of $V$ such that $v +3x = w$ .