Answer to Question #327705 in Linear Algebra for Mzalume

Question

Let 1 and 1 denote two distinct objects, neither of which is in R.

Define an addition and scalar multiplication on R [ f1g [ f1g as you

could guess from the notation. Specifically, the sum and product of two

real numbers is as usual, and for t 2 R define

t1 D

8

ˆ<

ˆ:

1 if t < 0;

0 if t D 0;

1 if t > 0;

t .1/ D

8

ˆ<

ˆ:

1 if t < 0;

0 if t D 0;

1 if t > 0;

t C1D1C t D 1; t C .1/ D .1/ C t D 1;

1C1D1; .1/ C .1/ D 1; 1 C .1/ D 0:

Is R [ f1g [ f1g a vector space over R? Explain.