High School

We appreciate your visit to Let tex l 1 tex and tex l 2 tex be operators on a finite dimensional vector space Prove that if tex l 1 tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Let [tex]l_1[/tex] and [tex]l_2[/tex] be operators on a finite-dimensional vector space. Prove that if [tex]l_1[/tex] is not invertible, then the product [tex]l_1 l_2[/tex] is not invertible.

Answer :

Final answer:

An operator on a finite-dimensional vector space is invertible if its null space is zero. Since l1 is not invertible, its null space isn't zero. Applying this to the product operator l1*l2 we find it's also not invertible.

Explanation:

In linear algebra, operators on a finite-dimensional vector space refer to linear transformations that map the vectors space to itself. To prove the statement, we can make use of the theorem which states that a linear operator is invertible if and only if its null space is zero.

Assume l1 is not invertible. This implies that its null space is not equal to zero. Therefore, there exists a non-zero vector v such that l1(v) = 0.

If we consider the operator l1*l2, we can test for its invertibility by following the same path. We find that l1*l2(v) = l1(0) = 0. So l1*l2 is also not invertible because its null space has a nonzero vector v.

This is the reasoning which proves that if l1 is not invertible then the product l1*l2 is not invertible as well.

Learn more about Linear operators on finite-dimensional vector space here:

https://brainly.com/question/33949789

#SPJ11

Thanks for taking the time to read Let tex l 1 tex and tex l 2 tex be operators on a finite dimensional vector space Prove that if tex l 1 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada