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HomogenousCow
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Can't quite see why a one-to-one linear transformation is also onto, anyone?
HomogenousCow said:Can't quite see why a one-to-one linear transformation is also onto, anyone?
HomogenousCow said:Can't quite see why a one-to-one linear transformation is also onto, anyone?
I guess I was assumming the same dimension for map, i.e., map from ##\mathbb R^n \rightarrow \mathbb R^n ## or any two vector spaces of the same dimension. There are other ways of seeing this. EDIT: Mayb be more accurate to say that map T is of full rank than saying it is onto.WWGD said:Rank Nullity Theorem: Nullity is zero...
Onto equivalence and one-to-one equivalence are two different ways to describe the same property of a linear transformation. Onto equivalence means that the transformation maps every element in the range to a unique element in the domain. One-to-one equivalence means that the transformation maps each element in the domain to a unique element in the range. Essentially, this means that the transformation is both injective and surjective.
The difference between onto equivalence and one-to-one equivalence lies in the perspective from which the transformation is viewed. Onto equivalence focuses on the range of the transformation, ensuring that every element in the range is mapped to a unique element in the domain. One-to-one equivalence, on the other hand, focuses on the domain of the transformation, ensuring that every element in the domain is mapped to a unique element in the range.
A linear transformation is invertible if and only if it is both onto and one-to-one. This means that the transformation has a unique inverse that maps the range back to the domain, and vice versa. This inverse can only exist if the transformation is onto equivalent to one-to-one, as it ensures that there are no elements in the range that are mapped to multiple elements in the domain.
No, a linear transformation cannot be onto equivalent to one-to-one without being invertible. As mentioned earlier, invertibility is a necessary condition for onto equivalence and one-to-one equivalence. If the transformation is not invertible, there must be either elements in the range that are not mapped to in the domain, or multiple elements in the range that are mapped to the same element in the domain, violating the requirements for onto equivalence and one-to-one equivalence.
To determine if a linear transformation is onto equivalent to one-to-one, you can use the rank-nullity theorem. This theorem states that for a linear transformation from a vector space of dimension n to a vector space of dimension m, if the rank of the transformation is equal to n (i.e. the transformation is onto) and the nullity of the transformation is equal to 0 (i.e. the transformation is one-to-one), then the transformation is onto equivalent to one-to-one. So, you can calculate the rank and nullity of the transformation and check if they satisfy these conditions.