Proving an image and annihilator of a kernel are equal

In summary, using the given assumptions, it can be proven that T:V→U is linear and V has finite dimension. It can be shown that I am Tt = (Ker T)0, and that dim((Ker T)0) = dim(I am Tt). This can be proven using a theorem that states that given linear mappings A:V→U and w:V→R, if the kernel of A is a subset of the kernel of w, then there exists a linear mapping Λ:U→R such that w=ΛA.
  • #1
Adgorn
130
18

Homework Statement


Suppose T:V→U is linear and V has finite dimension. Prove that I am Tt = (Ker T)0

Homework Equations


dim(W)+dim(W0)=dim(V) where W is a subspace of V and V has finite dimension.

The Attempt at a Solution


I first proved I am Tt ⊆ (Ker T)0. Let u be an arbitrary element of Ker T and Tt(Φ )∈ I am Tt, then Tt(Φ)(u)=Φ(T(u))=Φ(0)=0. Thus, since all elements of I am Tt map all elements of Ker T into 0, I am Tt ⊆ (Ker T)0.

I now only need to proof dim((Ker T)0) = dim I am Tt. Since V has finite dimension, dim((Ker T)0) = dim(V)-dim(Ker T), but dim(Im T) = dim(V)-dim(Ker T). Thus, dim((Ker T)0)=dim(Im T). This is where I got stuck, how do I prove dim (Im T) = dim ((Im T)0)?
 
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  • #2
Adgorn said:
V has finite dimension
this assumption is redundant. Redundant assumptions just obscure a core of the theorem and make proof more complicated than it really is.
 
Last edited:
  • #3
Hint.
First prove a
Theorem. Let ##A:V\to U,\quad w:V\to\mathbb{R}## be lenear mappings such that ##\mathrm{ker}\,A\subseteq \mathrm{ker}\,w##. Then there exist a linear mapping ##\Lambda:U\to\mathbb{R}## such that ##w=\Lambda A##.

It is actually a special case of some another general theorem
 

Related to Proving an image and annihilator of a kernel are equal

1. What is a kernel?

A kernel in mathematics is a set of elements that map to the identity element of another set under a given operation.

2. What is an image?

An image is a set of elements that are produced by applying a given operation to the elements of another set.

3. How do you prove that an image and an annihilator of a kernel are equal?

To prove that an image and an annihilator of a kernel are equal, you need to show that they have the same elements. This can be done by showing that any element in the image is also in the annihilator of the kernel, and vice versa.

4. Why is it important to prove that an image and an annihilator of a kernel are equal?

Proving that an image and an annihilator of a kernel are equal is important because it helps to establish a fundamental relationship between two mathematical concepts. It also allows for the simplification of certain operations and can aid in solving more complex mathematical problems.

5. Are there any real-life applications of proving an image and an annihilator of a kernel are equal?

Yes, there are many real-life applications of this concept, especially in the fields of computer science, engineering, and physics. For example, in computer graphics, the concept of an image and a kernel is used to manipulate and transform images. In physics, this concept is used to understand and solve problems related to wave propagation and signal processing.

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