Consider the circle in the xyplane centered at the origin wh

Consider the circle in the xy-plane centered at the origin whose equation is x^2 + y^2 = 1. Let W be the set of all vectors whose tail is at the origin and whose head is a point inside or on the circle. Is W a subspace of R^2? Explain.

Solution

No it is not a subspace

Consider two vectors in W

(1,0) and (0,1)

For W to be a subspace sum of these two vectors ie (1,1) must also be in W

But it is not

||(1,1||=sqrt{1+1}=sqrt{2}>1

Hence th point (1,1) is outside the unit circle hence not in W

HEnce W is not a subspace