Subset & Subspace Homework: Closed Under Vector Addition & Scalar Multiplication

[negation]

Homework Statement

a) Find a set of vectors in R2 that is closed under vector addition but not under scalar multiplication
Find a set of vectors closed under scalar multiplication but not closed under vector addition.

The Attempt at a Solution

a) Let S be a set of vectors in R2.

S = {(x,y) | x + y =0}
x = (1,1) y = (-1,-1)

To show that S set of vectors is closed under vector addition, x + y must remain in S.

x + y = (x1 + y1, x2+y2) = ( 0,0)

Am I right up till here?

 

[Mark44]

Homework Statement

a) Find a set of vectors in R2 that is closed under vector addition but not under scalar multiplication
Find a set of vectors closed under scalar multiplication but not closed under vector addition.

The Attempt at a Solution

a) Let S be a set of vectors in R2.

S = {(x,y) | x + y =0}
x = (1,1) y = (-1,-1)

To show that S set of vectors is closed under vector addition, x + y must remain in S.

x + y = (x1 + y1, x2+y2) = ( 0,0)

Am I right up till here?

No. Your set S is the line whose equation is x + y = 0. This is a line through the origin, and as such, this set is a one dimensional subspace of R². You need to find a different set of vectors.

Look at the examples in your book or notes. There are probably some examples of sets that are closed under one operation, but not the other.

Also note that this is two problems - one for a set that is closed under vector addition but not under multiplication by a scalar; the other is closed under scalar multiplication but not vector addition.