Let W be the subset of R3 consisting of all vectors x y z sa

Let W be the subset of R^3 consisting of all vectors (x, y, z) satisfying x = 5y. Is W a subspace of R^3? Let W be the subset of R^2 consisting of all vectors (x, y) with |x| + [y] = 1. Is W a subspace of R^2?

Solution

1) x = 5y

If (x1 , y1 , z1) + ( x2, y2 , z2 ) = (x 1+x2 , y1+y2 , z1 +z2)

x1 +x2 -5(y1+y2) =0

So, (x1 -5y1 ) + ( x2 - 5y2) =0

So, closed under addition

Check for closed under scalar multiplication :

( x, y , z) = (kx , ky , kz)

kx - 5ky =0

k( x -5y) =0

Closed under scalar multiplication

(0, 0, 0, ) is also a soultion of the system of equation

So, It is a subspace of R3

2) |x| + |y| = 1

( x1 , y1) + ( x2 , y2) = ( x1 +x2 , y1 +y2)

|x1 +x2 | + | y1 +y2| -1 =0

So, this is not equal to |x1 +y1| + |x2+ y2| -1

So, not closed under addition
Not a subspace of R2