Geometric intepretation of matrices

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Discussion Overview

The discussion centers on the geometric interpretation of matrices, particularly in the context of linear transformations within various vector spaces, including polynomial spaces and real vector spaces. Participants explore how to visualize these transformations and the challenges associated with higher-dimensional spaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that an n x n matrix can represent a linear transformation of an n-dimensional vector space.
  • One participant questions how to geometrically interpret a linear transformation from the vector space of polynomials up to degree 2 to that of degree 3.
  • Another participant emphasizes the importance of visualization in understanding linear algebra, contrasting it with their experience in calculus.
  • There is a suggestion that visualizing linear transformations in 1, 2, and 3 dimensions can be done by analogy for higher dimensions.
  • One participant notes the challenge of visualizing transformations from a 4 x 4 matrix space to polynomial spaces, citing the dimensionality of the spaces involved.
  • Another participant explains that the specific nature of elements in a vector space is not crucial for visualization, as dimensions can be treated similarly across different spaces.
  • A participant provides examples of linear transformations between polynomial spaces, such as integration and differentiation, suggesting that geometric interpretation can derive from calculus knowledge.
  • A link to an external resource is shared as a reference for further exploration.

Areas of Agreement / Disagreement

Participants express varying levels of comfort with visualizing linear transformations, particularly in higher dimensions. While some agree on the importance of visualization, others highlight the complexities involved, indicating that the discussion remains unresolved regarding the best methods for interpretation.

Contextual Notes

The discussion touches on limitations in visualizing higher-dimensional spaces and the dependence on the choice of basis vectors for interpretation. There is also an acknowledgment of the challenges in associating transformations with geometric representations.

JG89
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Is there a geometric interpretation of any n*n matrix?
 
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An n x n matrix can represent a linear transformation of an n dimensional vector space.
 
And how would I geometrically interpret a linear transformation from, say, P(2) to P(3) ?( P(n) is the vector space of polynomials up to the nth degree)
 
What do you mean by 'geometrically interpret'? Do you want to visualize it?
 
Yeah. I find that I'm usually better at Calculus than Linear Algebra because I am able to visualize it easily. Usually for linear algebra I don't know how to visualize things.
 
Do you know how to visualize a linear transformation of 1, 2 and 3 dimensional real vector spaces? You'll just have to think by analogy for higher dimensions.
 
If you mean a linear transformation mapping from R^n, where n goes from 1 to 3, then yeah, this is no problem for me. But say I had one from the vector space of 4*4 matrices to the vector space of polynomials up to the n'th degree. How would I be able to visualize this?
 
The vector space of 4 x 4 matrices is 16 dimensional, so it's not possible in the usual sense.

Also, the specific nature of the elements of the vector space is not relevant. For example, you can visualize the vector space of polynomials with real coefficients up to the second degree in the same you you visualize R3, because the two spaces have the same dimension. You can choose the basis vectors for the former as 1, x and x2, and treat them visually the same way you treat (1, 0, 0), (0, 1, 0) and (0, 0, 1).
 
There are two cases you can visualize without having to associate curves with points in with Euclidean n-space. One example of a linear transformation from P(2) into P(3) is an indefinite integral restricted to a 0 constant of integration. In reverse, one linear transformation from P(3) into P(2) is differentiation. You can even go ahead and prove these statements and find the matrix form of these two operators with respect to whatever basis you want. The geometric interpretation comes from your knowledge of calculus.
 
  • #10
Best of what I have seen on the net so far...

http://www.uwlax.edu/faculty/will/svd/action/index.html
 
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