Geometric intepretation of matrices

[JG89]

Is there a geometric interpretation of any n*n matrix?

 

[dx]

An n x n matrix can represent a linear transformation of an n dimensional vector space.

 

[JG89]

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)

 

[dx]

What do you mean by 'geometrically interpret'? Do you want to visualize it?

 

[JG89]

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.

 

[dx]

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.

 

[JG89]

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?

 

[dx]

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).

 

[slider142]

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.

 

[trambolin]

Best of what I have seen on the net so far...

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