Thread: Question involving Vector Spaces

Let V be the set of all ordered pairs of real numbers with addition defined by

(x1, x2) + (y1, y2) = (x1 + y1, x2 + y2)

and scalar multiplication defined by

α¤(x1, x2) = (αx1, x2)

The scalar multiplication for this system is defined in an unusual way, and consequently we use the symbol ¤ to avoid confusion with the ordinary scalar multiplication of vectors. Is V a vector space with these operations? Justify your answer.

I honestly would think not because α(x1, x2) should equal (αx1, αx2) however the problem states it is not ordinary scalar multiplication. It doesn't state what kind though . Help would be appreciated, thank you!

Hi

What is meant by "The scalar multiplication for this system is defined in an unusual way" is that you do not use (which is the classical product) but , that's all. The product is the usual product of real numbers.

Originally Posted by pakman

Let V be the set of all ordered pairs of real numbers with addition defined by

(x1, x2) + (y1, y2) = (x1 + y1, x2 + y2)

and scalar multiplication defined by

α¤(x1, x2) = (αx1, x2)

The scalar multiplication for this system is defined in an unusual way, and consequently we use the symbol ¤ to avoid confusion with the ordinary scalar multiplication of vectors. Is V a vector space with these operations? Justify your answer.

...

I assume that you know the axioms of a vector space.

You only have to check if all (8) axioms are satisfied by the given definitions.