# Thread: Question involving Vector Spaces

1. ## 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!

2. Hi

What is meant by "The scalar multiplication for this system is defined in an unusual way" is that you do not use $\displaystyle \alpha(x_1,x_2)=(\alpha x_1,\alpha x_2)$ (which is the classical product) but $\displaystyle \alpha(x_1,x_2)=(\alpha x_1,x_2)$, that's all. The product $\displaystyle \alpha x_1$ is the usual product of real numbers.

3. 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.