Let V-R^2 with the standard scalar multiplication and addition + · defined by: (x1,y1)+·(x2,y2)=(x1+x2,y1-y2). Show that this is NOT a vector space.

lfryerda | High School Teacher | (Level 2) Educator

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Since each element `v in R^2` has standard scalar multiplication, to prove that this is not a vector space, we need to focus on the vector space axioms that relate to the vector space axioms of addition.

To prove that this is not a vector space, we just need to find a single example where one of the rules fails.  In particular, consider the rule for associativity:

`(u+v)+w=u+(v+w)`

and consider the three vectors:

`u=(0,1)`

`v=(0,2)`

`w=(0,3)`

Now the left side of the associativity rule gives:

`(u+v)+w`

`=((0+0)+0,(1-2)-3)`

`=(0,-1-3)`

`=(0,-4)`

On the other hand, the right side of the associativity rule gives:

`(0+(0+0),1-(2-3))`

`=(0,1-(-1))`

`=(0,2)`

Since the two sides are not equal, then the associativity rule for vector spaces fails.

This is not a vector space.