If V is a vector space, then prove that if c.u=0, then either c=0 or u=0.?

If V is a vector space, then prove that if c.u=0, then either c=0 or u=0.


Here, c is a scalar and u is a vector.


Try to prove the contrapositive, i.e., if c and u are non-zero, then c.u is non-zero.


Here the '+' and '.' are the vector addition and scalar multiplication operators respectively.

If V is a vector space, then prove that if c.u=0, then either c=0 or u=0.?
suppose c is non-zero


In cu=0, multiply with the inverse of c





c^(-1)cu=0 which is (c^(-1)c)u=0 or 1u=0





Then try to use one of the properties in the definition of vectorial space

floral