Vector transformations that lead to the identity matrix

In summary, a vector transformation that leads to the identity matrix is when a vector is multiplied by a matrix and the resulting vector is equal to the original vector. This is important to understand because it allows for easy manipulation and analysis of vectors and matrices, and serves as the basis for many mathematical concepts and calculations. Vector transformations can be represented geometrically using vectors and their corresponding matrices, and can be reversed by multiplying by the inverse of the transformation matrix. Real-world applications of these transformations include computer graphics, robotics, physics, engineering, navigation systems, data compression, and encryption algorithms.
  • #1
geert200
1
0
Hi all,

I have a question that seems very simple but I just do not see it;)

Let α denote an r×1 vector with arbitrary entries; I'm trying to construct an 1×r vector m such that αm = I, where I is the r×r identity matrix...

The first question is: is this possible?

I tried the following;

let m = α'(α α')^{-1}, but then the problem is that (α α')^{-1} is not defined (rank 1)
how can I fix this;

Thanks in advance Geert
 
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  • #2
You can't. For any two vectors α and m, αm has rank 1. I has rank r.
 

Related to Vector transformations that lead to the identity matrix

1. What is a vector transformation that leads to the identity matrix?

A vector transformation that leads to the identity matrix is when a vector is multiplied by a matrix and the resulting vector is equal to the original vector. In other words, the matrix has no effect on the vector and the resulting transformation is equivalent to the identity matrix.

2. Why is it important to understand vector transformations that lead to the identity matrix?

Understanding vector transformations that lead to the identity matrix is important because it allows us to easily manipulate and analyze vectors and matrices. It also serves as the basis for many mathematical concepts and calculations, such as finding the inverse of a matrix.

3. How can vector transformations be represented geometrically?

Vector transformations can be represented geometrically using vectors and their corresponding matrices. Each vector represents a point in space, and when multiplied by a matrix, it is transformed into a new point. The resulting transformation can be visualized as a rotation, scaling, reflection, or shearing of the original vector.

4. Can vector transformations that lead to the identity matrix be reversed?

Yes, vector transformations that lead to the identity matrix can be reversed. This is because the identity matrix is its own inverse. This means that when a vector is multiplied by the inverse of the transformation matrix, it will return to its original position.

5. What are some real-world applications of vector transformations that lead to the identity matrix?

Vector transformations that lead to the identity matrix have many real-world applications, including computer graphics, robotics, and physics. They are also used in engineering and navigation systems for transformations such as scaling and rotating objects. Additionally, they play a crucial role in data compression and encryption algorithms.

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