MHB Matrix Transforms: nxm, n->m, m->n, n+m->n/m

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Matrix transformations involve converting vectors between different dimensions, specifically from nxm to m and vice versa. A matrix of dimension nxm can transform a vector of dimension n into a vector of dimension m and also handle transformations from m to n. Additionally, it can convert vectors of dimension n+m into either m or n. The discussion emphasizes the importance of understanding matrix multiplication rules, particularly how dimensions interact during these transformations. Clarity on the number of rows and columns in a matrix is crucial for accurate application of these concepts.
Imiebee
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A matrix of dimension nxm
a. transforms a vector of dimension n to a vector of dimension m
b. transforms a vector of dimension m to a vector of dimension n
c. a vector of dimension n+m to a vector of dimension m
d. a vector of dimension n+m to a vector of dimension n
 
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Matrix multiplication (note that a vector is just an m x 1 matrix)

(n x m) X (m x k) => (n x k)

See if you can figure it out now.
 
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Imiebee said:
A matrix of dimension nxm
So which is the number of rows and which is the number of columns?

a. transforms a vector of dimension n to a vector of dimension m
b. transforms a vector of dimension m to a vector of dimension n
c. a vector of dimension n+m to a vector of dimension m
d. a vector of dimension n+m to a vector of dimension n
 
I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...

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