There is only one linear transformation which maps between bases of equal size

up:: Linear Transformation

Given Vector Spaces with Hamel Basis . Then there is a unique linear transformation such that

each element in gets mapped to .

Proof

Since is a basis for , then each vector can be written as a Linear Combination

If we seek a linear transformation, it must preserve these combinations, as well as the hypothesis that .

Therefore, define such that

is linear

Having with respective linear combinations , we have that

whose image under is

is unique

Let be a linear transformation which also preserves basis elements .

Let . Then

Thus, , and thus .

Corollaries

• Every vector space with same dimension are isomorphic to each other

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References

• Um Curso de Álgebra Linear, Flávio Ulhoa Coelho & Mary Lilian Lourenço. Editora EDUSP.