Line Image Transformation with 3x3 Matrix - Vector Solution

In summary, the conversation discusses finding the images of lines under a transformation using a given matrix. The image is found by multiplying the matrix by the vector representing the line. The use of an inverse matrix is necessary to find the "inverse image" of a given vector.
  • #1
takercena
25
0
Please help me solve this question:
Question: Find the images of the lines r = i - j + n(i + j + k) under transformation
M (matrix 3 x 3) =
(2 1 4)
(3 5 1)
(1 2 0)
in vector form.
 
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  • #2
The image under the transformation is just that matrix times the vector? But what is n? My first thought was that it was the parameter but then that is just a single line. If that is correct then r= i- j+ n(i+ j+ k)= (n+1)i+ (n-1)j+ nk and its image is
[tex]\left[\begin{array}{ccc}2 & 1 & 4 \\ 3 & 5 & 1 \\ 1 & 2 & 0\end{array}\right]\left[\begin{array}{c} n+1 \\ n-1 \\ n\end{array}\right][/tex]
 
  • #3
Thank you. There is some confusion here. I am confuse when to use inverse matrix to find transformation. Can you explain this. Example : 1. Find the images under the transformation
 
Last edited:
  • #4
I have no idea what you mean by that. You use the inverse matrix in order to find an "inverse image": x such that Mx= y. Given a matrix M, the image of x is, by definition Mx.
 

1. What is the "Transformation problem"?

The "Transformation problem" refers to a theoretical issue in economics that arises when attempting to transform the values of commodities (goods and services) into prices. It is a central debate in the study of Marxist economics.

2. Why is the "Transformation problem" important?

The "Transformation problem" is important because it challenges the traditional understanding of how prices are determined in a capitalist economy. It also has implications for the labor theory of value and the concept of exploitation in Marxist theory.

3. What are some proposed solutions to the "Transformation problem"?

There are several proposed solutions to the "Transformation problem", including the simultaneous solution, the temporal single-system interpretation, and the dual-system interpretation. Each solution offers a different perspective on how to reconcile the values of commodities with their prices.

4. How does the "Transformation problem" relate to capitalist economies?

The "Transformation problem" is relevant to capitalist economies because it challenges the neoclassical theory of prices, which argues that prices are determined by the intersection of supply and demand. It also highlights the role of labor and exploitation in the capitalist system.

5. Can the "Transformation problem" be solved?

The "Transformation problem" is an ongoing debate and has not been definitively solved. Some economists argue that it is a theoretical problem that does not have practical implications, while others continue to search for a solution. Ultimately, the answer to this question depends on one's perspective and interpretation of the problem.

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