LINEAR ALGEBRA Please, label all steps/solutions clearly and concisely. Thank yo

Question

LINEAR ALGEBRA

Please, label all steps/solutions clearly and concisely.
Thank you.

#7.1

I. Let V R2 and consider the following subsets of V . (a) Let S be the subset where 12.21 S if xiT2 = 0. Show that S is closed under scalar multiplication and has the additive identity, but is not closed under addition. (b) Let S be the subset wheif ri and r2 are integers. Show that S is closed under addition and has the additive identity, but is not closed under scalar multiplication (c) Let S consist of no elements at all. Show that S is closed un der addition and scalar multiplication (think about the definitions carefully!), but does not have the additive identity.

Explanation / Answer

a.

Let, (x,y) be in S and c a scalar

So, (cx)(cy)=c^2(xy)=0

Hence, c(x,y)=(cx,cy) is in S hence closed under scalar multiplication

Identity element is the 0 vector

Consider the elements

(0,1),(1,0) both in S

Adding them gives

(1,1) which is not in S

b)

The additive identity is the 0 vector which is in S

Let, (x,y),(u,v) be in S so that x,y,u,v are integers

(x,y)+(u,v)=(x+u,y+v) is in S as

x+u,y+v are also integres

Now consider

(1,1) in S and scalar 0.5

Multipplying gives

(0.5,0.5) which is not in S

c)

Clearly there is no additive identity as there are no elements in S

For any element not in S it cannot be written as sum of two element in S as S has no elements

Hence, S is closed under addition

For any elements not in S it cannot be written as scalar multiplie of element in S as there are no element in S

HEnce closder under scalar multiplication

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