Exploring General Linear Transformations of p Vectors in R(n) and R(m)

In summary, the conversation discusses the concept of a general linear transformation of p vectors in R(n) to R(m). The main point of confusion is the meaning of "p vectors", with the questioner clarifying that it refers to p number of vectors in R(n). The questioner also asks for a matrix that can perform this transformation, possibly by using v_i as column vectors.
  • #1
vish22
34
1
Ok just for fun,could someone please give a general linear transformation of p vectors in R(n) to R(m),by expressing the transformation as a Matrix vector product of let's say n vectors in R(m).p vectors in R(n).I've already done it for fun but I'd like to see how you guys go about it..
 
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  • #2
When you say "a general linear transformation of p vectors", what do you mean by "p vectors"?
 
  • #3
as in "p" no. of vectors in R(n)
 
  • #4
Your problem is not worded properly. So we have [itex]v_1, \ldots, v_p \in \mathbb{R}^n[/itex] and you want a matrix that does what?

I really don't understand the point of the p. Do you want a matrix with [itex]v_i[/itex] as column vectors, and apply a GLT to the resulting matrix?
 
  • #5
yea,the 1st part is right.. v1,v2...vp vectors in N-space-a GLT of these vectors expressed as a product with N vectors (represented in a matrix ofc) in M-space!
 
  • #6
I don't consider trying to guess what you mean to be "fun".
 

Related to Exploring General Linear Transformations of p Vectors in R(n) and R(m)

1. What is the purpose of exploring general linear transformations?

The purpose of exploring general linear transformations is to better understand how different transformations can affect a vector or set of vectors in a multivariate space. This can help in various fields such as mathematics, physics, and computer science.

2. How do you define a general linear transformation?

A general linear transformation is a mathematical function that maps a vector or set of vectors from one vector space to another, while preserving the fundamental properties of the vector space such as linearity and preserving operations like addition and scalar multiplication.

3. What is the difference between R(n) and R(m) in the context of linear transformations?

R(n) and R(m) refer to n-dimensional and m-dimensional vector spaces, respectively. In the context of linear transformations, R(n) and R(m) represent the domain and codomain of the transformation, respectively. For example, a transformation from R(3) to R(2) would take in a 3-dimensional vector and output a 2-dimensional vector.

4. What are some common applications of general linear transformations?

General linear transformations have numerous applications in various fields. Some common applications include image and signal processing, data compression, computer graphics, and machine learning.

5. How can one determine the effect of a general linear transformation on a vector or set of vectors?

To determine the effect of a general linear transformation, one can use the transformation matrix associated with the transformation. The transformation matrix contains information about how the transformation affects the basis vectors in the vector space. By multiplying the transformation matrix with a given vector or set of vectors, one can observe the resulting transformation.

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