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.

 

[blue_raver22]

Vector addition is a fundamental operation in linear algebra, which is the branch of mathematics that deals with vector spaces and their properties. It is defined as the process of combining two or more vectors to obtain a new vector. In the context of multivariable calculus, vector addition is typically denoted by the symbol "+", and it follows the same rules as addition of real numbers.

The notation V \times V \rightarrow V may seem unfamiliar, but it is simply a way of representing the process of vector addition. The "V \times V" part represents the Cartesian product of two sets, which in this case are both the set of vectors denoted by V. The Cartesian product is a way of combining elements from two sets to form ordered pairs. In the context of vector addition, it represents the combination of two vectors to obtain a new vector.

The " \rightarrow V" part indicates that the result of the vector addition operation is a vector, which belongs to the same set as the original vectors. This is important because vector addition is a closed operation, meaning that the result will always be a vector in the same vector space as the original vectors.

In summary, the notation V \times V \rightarrow V represents the process of combining two vectors from the set V to obtain a new vector in the same set V. This is the essence of vector addition, which is a fundamental operation in linear algebra and plays a crucial role in many mathematical and scientific applications.