Proving Existence of Linear Mapping with Kernel in Subspace S | Helpful Guide

In summary, The theorem states that for any subspace S in a finite dimensional vector space V, there exists a Linear Mapping L: V → V whose kernel is S. This can be proved by taking the orthogonal projection to the complement subspace of S.
  • #1
Bazzinga
45
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Hey guys, I was wondering if you could help me out with a question I've got, I really don't know where to go or really where to start! Here's the question:

Let S be a subspace of a finite dimensional vector space V. Show that there exists a Linear Mapping L: V → V such that the kernel of L is S.

I started off messing around with some examples and the theorem makes sense to me, I just can't figure out how to prove it! If someone could start me off that would be awesome.

Thanks!
 
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  • #2
Take the orthogonal projection to the complement subspace of S.
 

FAQ: Proving Existence of Linear Mapping with Kernel in Subspace S | Helpful Guide

What is a tricky linear mapping proof?

A tricky linear mapping proof is a mathematical proof that involves demonstrating the properties and relationships of linear transformations between vector spaces. These proofs often involve complex algebraic manipulations and can be challenging to solve.

Why are linear mapping proofs important?

Linear mapping proofs are important because they help us understand the behavior and properties of linear transformations, which are used in many areas of mathematics and science. These proofs also have practical applications in fields such as engineering, physics, and computer science.

What are some common techniques used in tricky linear mapping proofs?

Some common techniques used in tricky linear mapping proofs include using properties of matrices, applying theorems such as the rank-nullity theorem and the invertible matrix theorem, and using algebraic manipulations to simplify equations.

What are some common challenges in solving tricky linear mapping proofs?

Some common challenges in solving tricky linear mapping proofs include understanding the properties and definitions of linear transformations, manipulating complex equations, and recognizing how different techniques and theorems can be applied to a specific proof.

How can I improve my skills in solving tricky linear mapping proofs?

To improve your skills in solving tricky linear mapping proofs, it is important to have a strong foundation in linear algebra and to practice solving different types of proofs. It can also be helpful to work with a study group or seek assistance from a teacher or tutor when encountering challenging proofs.

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