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
Related Book For
Differential Equations and Linear Algebra
ISBN: 978-0131860612
2nd edition
Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West
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