Definition of vector addition, Cartesian product?

[vanmaiden]

I'm reading through a multivariable calculus book and it starts off with some linear algebra. It defines vector addition as V $\times$ V $\rightarrow$ V. My text describes V as a set and describes the above process as a mapping. I believe the $\times$ may represent a Cartesian product. Could someone fill me in on how such an operation could define vector addition?

 

[lavinia]

I'm reading through a multivariable calculus book and it starts off with some linear algebra. It defines vector addition as V $\times$ V $\rightarrow$ V. My text describes V as a set and describes the above process as a mapping. I believe the $\times$ may represent a Cartesian product. Could someone fill me in on how such an operation could define vector addition?

Vector addition just takes two vectors and gives you third. So the map is

(X,Y) -> X + Y.

 

[algebrat]

I'm reading through a multivariable calculus book and it starts off with some linear algebra. It defines vector addition as V $\times$ V $\rightarrow$ V. My text describes V as a set and describes the above process as a mapping. I believe the $\times$ may represent a Cartesian product. Could someone fill me in on how such an operation could define vector addition?

It's a product of sets, which we map out of with addition. Yes, it is the Cartesian product, in the sense that Cartesian product is product of sets. You can map any point in the product (x,y), as lavinia said, to x+y for instance.

You might contrast product and/or sum of sets with product and/or sum of elements.

 

[jcsd]

You could do the same for normal addition (i.e. the addition of the real numbers).

I.e. you define a function +:ℝ²→ℝ (where ℝ²=ℝ×ℝ)

For example +(13,2) = 15

Infact +:ℝ²→ℝ is actually an example of vector addition as the reals themselves form a 1-D real vector space wrt real addition and real multiplication.