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
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).
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.
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.
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.
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.