Question

can someone show that the set of fourth quadrant vectors is not a subspace of R^2(2- dimension)

Answer 1

Consider some scalar \(c\in\mathbb{R}, c<0\), then consider any vector \(\vec v\in \mathbb{R}^2, |\vec v|>0\) in the fourth quadrant (which we shall denote \(\mathbb{R}^2_{\text{IV}}\)), then \(c\vec v \not \in \mathbb{R}^2_{\text{IV}}\). Since \(\mathbb{R}^2_{\text{IV}}\) is not closed under scalar multiplication, \(\mathbb{R}^2_{\text{IV}}\) is not a subspace of \(\mathbb{R}^2\).
Q.E.D.