Examples
- Consider R2, the vector space of all coordinates (a, b) where both a and b are real numbers. Then a very natural and simple basis is simply the vectors e1 = (1,0) and e2 = (0,1): suppose that v = (a, b) is a vector in R2, then v = a (1,0) + b (0,1). But any two linearly independent vectors, like (1,1) and (−1,2), will also form a basis of R2.
- More generally, the vectors e1, e2, ..., en are linearly independent and generate Rn. Therefore, they form a basis for Rn and the dimension of Rn is n. This basis is called the standard basis.
- Let V be the real vector space generated by the functions et and e2t. These two functions are linearly independent, so they form a basis for V.
- Let R denote the vector space of real polynomials; then (1, x, x2, ...) is a basis of R. The dimension of R is therefore equal to aleph-0.
Read more about this topic: Basis (linear Algebra)
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