4. For o E Sn, define a matrix U := [di,o(j)]i,j=1,...,n M(R) Note that &...

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4. For o E Sn, define a matrix U₁ := [di,o(j)]i,j=1,...,n € M₂(R) Note that & denotes the Kronecker delta function: [1, i =j; 0, i ‡j. di,j = (a) Prove that U¿U₁ = Uμ for σ, TE Sn. T στ (b) Let R be a unital ring. Prove that the formula * ( Σ00) - Σ := oESn σεβη defines a homomorphism from R.Sn to Mn(R). Σ ao Uo 4. For o E Sn, define a matrix U₁ := [di,o(j)]i,j=1,...,n € M₂(R) Note that & denotes the Kronecker delta function: [1, i =j; 0, i ‡j. di,j = (a) Prove that U¿U₁ = Uμ for σ, TE Sn. T στ (b) Let R be a unital ring. Prove that the formula * ( Σ00) - Σ := oESn σεβη defines a homomorphism from R.Sn to Mn(R). Σ ao Uo

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a Let Sn Then the i jentry of UU is given by k1n ikkj ell1n kellellk ij On the other hand the i ... View the full answer