Prove that the standard basis vectors span R^2

In summary, the question is why we consider R^2 as a vector space over R instead of Q or any other field, and the difficulty of writing down a basis for R^2 over Q due to its uncountably infinite nature and the use of the axiom of choice.
  • #1
Mathematicsresear
66
0

Homework Statement


I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?

Standard basis vectors: e_1, e_2 or i,j
 
Physics news on Phys.org
  • #2
Mathematicsresear said:

Homework Statement


I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?

Standard basis vectors: e_1, e_2 or i,j

You will have difficulty writing down a basis of ##\mathbb{R}^2## over the field ##\mathbb{Q}##: such a basis is uncountably infinite and you need the axiom of choice to show it exists.
 

Related to Prove that the standard basis vectors span R^2

What does it mean for the standard basis vectors to span R^2?

When we say that the standard basis vectors span R^2, it means that any vector in R^2 can be written as a linear combination of the standard basis vectors, which are (1,0) and (0,1).

How can we prove that the standard basis vectors span R^2?

We can prove this by showing that any vector (a,b) in R^2 can be written as a linear combination of (1,0) and (0,1). We can do this by multiplying each basis vector by a scalar and adding them together, such that a(1,0) + b(0,1) = (a,b). This shows that the standard basis vectors span R^2.

Why is it important for the standard basis vectors to span R^2?

It is important because it means that we can represent any vector in R^2 in terms of the standard basis vectors. This property is essential in many mathematical and scientific applications.

Can any other set of vectors span R^2 besides the standard basis vectors?

No, the standard basis vectors are unique in their ability to span R^2. Any other set of vectors would not be able to represent all possible vectors in R^2.

Can we extend this concept to higher dimensions, such as proving that the standard basis vectors span R^3?

Yes, the concept of spanning can be extended to higher dimensions. In the case of R^3, the standard basis vectors are (1,0,0), (0,1,0), and (0,0,1), and we can show that any vector in R^3 can be written as a linear combination of these three vectors.

Similar threads

  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
2K
  • Calculus and Beyond Homework Help
Replies
16
Views
2K
  • Linear and Abstract Algebra
Replies
9
Views
647
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
9
Views
962
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Back
Top