Movement of a vector when multiplied by a matrix

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Multiplying a vector by a matrix transforms its direction and magnitude based on the properties of the matrix. For a given matrix, vectors may be scaled or rotated, with eigenvectors maintaining their direction while being scaled by their corresponding eigenvalues. The specific effects depend on the matrix's characteristics, such as whether it is singular or invertible. For example, multiplying vectors by the matrix [[1, 1], [1, 1]] results in all vectors being projected onto a line, effectively collapsing them into a single direction. Exploring various examples can further illustrate these transformations and their outcomes.
Ali Asadullah
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What is the effect on a vector when it is multiplied by a matrix?
Let any matrix
2 3
3 5
What will be the effect on vectors when they are multiplied with this matrix?
In which direction will they move?
What will be the effect of multiplying vectors with any matrix of order n by n?
 
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Different vectors are affected differently when multiplied by a given matrix. If the vector being multiplied by the matrix happens to be an eigenvector, the result vector is a vector in the same (or opposite) direction, scaled by the value of the eigenvalue associated with this eigenvector.
 
Let a matrix
1 1
1 1
If we multiply all the points/vectors in a plane with this matrix, what will be the resultant?
 
Ali Asadullah said:
Let a matrix
1 1
1 1
If we multiply all the points/vectors in a plane with this matrix, what will be the resultant?

What happens if you try some examples? Can you show us a few examples that you have worked out?
 

Thread 'Confusion about the Moyal-Weyl twist'

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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