What is the Corollary of the Nucleus and Image Theorem?

In summary, the conversation discusses the verification of three cases for a linear transformation, where case (a) is verified for n = 2, case (b) for n = 1, and case (c) for n = 0. The conversation also mentions the importance of showing that each case is possible for every n ≥ 2. The possibility of F(u) and F(v) being linearly independent or not is also discussed.
  • #1
Portuga
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TL;DR Summary
Consider ##u, v \in \mathbb{R}^2## such that ##\left\{ u,v \right\}## is a ##\mathbb{R}^2## basis. If ##F: \mathbb{R}^2 \rightarrow \mathbb{R}^n## a linear transformation, show that one of the following alternatives verifies:

(a) ##\left\{ F(u), F(v) \right\}## is linearly independent. (b) ##\dim \text{Im} (F) = 1## (c) ## \text{Im} (F) = {o}##
I tried hard to understand what this author proposed, but I feel like I failed miserably. My attempt of solution is here:
Item (a) is verified in the case where ##n = 2##, since ##F## being a linear transformation, by the Corollary of the Nucleus and Image Theorem, ##F## takes a basis of ##\mathbb{R}^2## to a ##\mathbb{R}^2## basis. Thus, ## \left \{ F (u), F (v) \right\}## is a basis of ##\mathbb{R}^2##.
Item (b) occurs in the case where ##n = 1##, because ##\dim \mathbb{R}^2 = 2, \dim \mathbb{R} = 1##, and by the Nucleus and Image theorem,
$$
\begin{align*}

& \dim\mathbb{R}^{2}=\dim\ker\left(F\right)+\dim\mathbb{R}\\

\Rightarrow & 2=\dim\ker\left(F\right)+1\\

\Rightarrow & \dim\ker\left(F\right)=2-1=1.

\end{align*}
$$
Item (c) is verified in case ##n = 0##.
I am pretty sure that I am very far away of the author's intention with this exercise, so please, any help would be appreciated.
 
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  • #2
The problem is not to find an [itex]n[/itex] for which each case is possible, but to show that each case is possible for every [itex]n \geq 2[/itex] (With [itex]n = 1[/itex] the first is impossible, since there is at most one linearly independent vector in [itex]\mathbb{R}^1[/itex].)

There are clearly two possibilities for [itex]n \geq 2[/itex]: Either [itex]F(u)[/itex] and [itex]F(v)[/itex] are linearly independent or they are not. If they are, we have case (1).

So suppose they are not linearly independent. Then there exist non-zero scalars [itex]A[/itex] and [itex]B[/itex] such that [tex]0 = AF(u) + BF(v)[/tex]. What can you say about [itex]Au + Bv[/itex] in this case?

The posbbility that [itex]F(u) = F(v) = 0[/itex] is not excluded.
 
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  • #3
Thank you very much! Now it's clear!
 
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Related to What is the Corollary of the Nucleus and Image Theorem?

1. What is a linear transformation problem?

A linear transformation problem involves transforming a set of data points in a linear fashion. This can include scaling, rotating, shearing, or reflecting the data points.

2. What is the purpose of solving a linear transformation problem?

The purpose of solving a linear transformation problem is to understand the relationship between different sets of data points and to make predictions or draw conclusions based on this relationship.

3. What are some real-world applications of linear transformation problems?

Linear transformation problems have many applications in fields such as engineering, physics, economics, and computer graphics. They can be used to model and analyze data in a variety of situations, such as predicting stock market trends, designing efficient building structures, and creating computer animations.

4. How do you solve a linear transformation problem?

To solve a linear transformation problem, you first need to determine the transformation matrix, which describes how the data points are being transformed. Then, you can apply this matrix to the original data points to obtain the transformed data points. Finally, you can analyze the transformed data to understand the relationship between the original and transformed data sets.

5. What are some common challenges when solving linear transformation problems?

One common challenge when solving linear transformation problems is determining the correct transformation matrix, as there can be multiple ways to transform the data points. Another challenge is interpreting and analyzing the transformed data, as it may not always be intuitive to understand the relationship between the original and transformed data sets.

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