The subset HC M(C) defined since it is closed under addition and multiplication, is in fact...

  

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The subset HC M₂(C) defined since it is closed under addition and multiplication, is in fact a subring. (a) Find a matrix for the inverse of x writ- ing for convenience |x|² := az + a² + a² + az. It should not look complicated. (This shows that every nonzero element of H is invertible, making H a division ring.) 1 i (b) Next, writing ( −1 0 0 0 1 -√) - ¹ - ( 1₁ ), * - ( √ ). A). j= k = -1 0 0 so that all x E H can be written x = ao + a₁i + a2j + a3k (where ao means aºÃ₂), what is x in these terms? (c) Derive a multiplication table for i, j, k (e.g., what is j², i . k, etc., in terms of linear combinations of 1 I2, i, j, and k with real coefficients). For example, i² = -1 and so via ao + a₁i you can think of C as lying inside of H. (Congratulations! It took Hamilton 10 years to produce this multiplication table, extending the complex numbers.) - The subset HC M₂(C) defined since it is closed under addition and multiplication, is in fact a subring. (a) Find a matrix for the inverse of x writ- ing for convenience |x|² := az + a² + a² + az. It should not look complicated. (This shows that every nonzero element of H is invertible, making H a division ring.) 1 i (b) Next, writing ( −1 0 0 0 1 -√) - ¹ - ( 1₁ ), * - ( √ ). A). j= k = -1 0 0 so that all x E H can be written x = ao + a₁i + a2j + a3k (where ao means aºÃ₂), what is x in these terms? (c) Derive a multiplication table for i, j, k (e.g., what is j², i . k, etc., in terms of linear combinations of 1 I2, i, j, and k with real coefficients). For example, i² = -1 and so via ao + a₁i you can think of C as lying inside of H. (Congratulations! It took Hamilton 10 years to produce this multiplication table, extending the complex numbers.) -

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The subset IHC M2C which consists of 2x2 matrices with complex entries is closed under addition and ... View the full answer