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Suppose \( E \) and \( F \) are vector spaces and \( f: E \rightarrow F \) is a linear map. Only one of the following four statements is true. Find the correct statement:

A) If \( f \) is one-to-one, then \( f \) is also onto.

B) If \( f \) is onto, then \( f \) is also one-to-one.

C) \( f \) is one-to-one if and only if \( f \) is onto.

D) If \( E \) and \( F \) have the same dimension, then \( f \) is one-to-one if and only if \( f \) is onto.

Answer :

The correct statement is c) "If E and F have the same dimension, then f is one to one if and only if f is onto."

A linear map (or linear transformation) f: E -> F between vector spaces E and F is said to be one-to-one (injective) if every element in E maps to a distinct element in F. On the other hand, a linear map f is said to be onto (surjective) if every element in F has a pre-image in E.

Statement (a) is not necessarily true. If f is one-to-one, it means that different elements in E are mapped to different elements in F. However, this does not guarantee that every element in F is reached. Therefore, if f is one-to-one, it does not imply that f is onto.

Statement (b) is also not necessarily true. If f is onto, it means that every element in F has a pre-image in E. However, this does not guarantee that different elements in E are mapped to different elements in F. Therefore, if f is onto, it does not imply that f is one-to-one.

Statement (c) is the correct statement. If E and F have the same dimension, it means that they have the same number of linearly independent vectors. In this case, if f is one-to-one, it implies that different elements in E are mapped to different elements in F. Additionally, if f is onto, it implies that every element in F has a pre-image in E. Therefore, if E and F have the same dimension, f is one-to-one if and only if f is onto.

To know more about linear maps, refer here:

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