High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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:

https://brainly.com/question/31944828#

#SPJ11

Other Questions