Assume that V and W are both vector spaces over R and that L: V notequaltorighta

Question

Assume that V and W are both vector spaces over R and that L: V notequaltorightarrow W is a linear map. Define the set N = {upsilon Element V: L(upsilon) = 0} Subsetequalto V. Prove that N is itself another vector space. Please state that you understand that N Subsetequalto V, and that you will accordingly use definition 5.4 and proposition 5.5 instead of checking all 10 properties in definition 5.1. Suppose L: V rightarrow W is a linear map. Prove that L is injective if and only if {upsilon Element V: L upsilon = 0} = {0}.

Explanation / Answer

Ans(3):

To prove N is vector space we need to show that N satisfies following two properties:
property1: If x belongs to N then ax belongs to N for some real number a.
Property2: If x,y belongs to N then (x+y) belongs to N

check for property 1:
Let x belongs to N then x belongs to V (by given definition of N)
Hence L(x)=0
similarly if a is some real number then ax belongs to V as V is a vector space
Hence L(ax)=aL(x)=a*0=0 (since L is linear map)
which means ax belongs N.

check for property 2:
Let x,y belongs to N then x,y belongs to V (by given definition of N)
Hence L(x)=0 and L(y)=0
now L(x+y)=L(x)+L(y)=0+0=0 (since L is linear map)

As both properties are satisfied by N hence N is a vector space.

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