Matrix Theory Determine whether the given subset U of R2 is

Matrix Theory

Determine whether the given subset U of R^2 is a subspace. Indicate (shade) the subset U in the plane. In case of failure, list the condition(s) that are false.

a) {(x,y) | xy >= 0}

b) {(x,y) | x = y}

c) {(x,y) | x =y^2}

Solution

a)

Not a subspace

Consider two points in U

(1,4) and (-2,-3)

Both are in U

Sum of the two points is

(1,4)+(-2,-3)=(-1,1) which is not in U

So, U is not closed under adition .HEnce not a subspace

b)

1. Let, (x,y) and (r,s) be in U

(x,y)+(r,s)=(x+r,y+s)

x+r=y+s

Hence, U is closed under addtiion

2. Let, (x,y) be in U and c be a scalar

cx=cy

Hece, c(x,y)=(cx,cy) is in U

Hence, U is closd under scalar multiplication

Hence, U is a subspace

c)

Not a subspace

Let, (1,1) and (4,2) be two points in U

Adding them gives

(5,3) which is not in U

So, U is not closed under addition

Hence, U is not a subspace