Group representation of S3 as six 2x2 matrixes.

Question

How to represent S3 as a 2x2 matrix. What does a basis e1-e2, e2-e3 mean? 1,2,3 are subscripts.

Answer 1

Hi there!

In linear algebra, e₁, e₂, e₃, …, e_n are standard unit vectors in a vector space Rⁿ, where their subscripts correspond to that entry in the vector that's 1.

That is to say, if you're working with 3D vectors in R³, then you would have standard unit vectors

e₁ = (1, 0, 0); e₂ = (0, 1, 0); e₃ = (0, 0, 1). If you write these in matrix notation, you would have

1

e₁ = 0

0

0

e₂ = 1

0

0

e₃ = 0

1

And, as the basis vectors for a 3x3 matrix, you would get the identity matrix I₃:

1 0 0

0 1 0

0 0 1

I hope this helped, and let me know if you have any more questions!