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Five.I Complex Vector Spaces Linear Algebra Jim Hefferon
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Chapter Five. Similarity
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Scalars will now be complex This chapter requires that we factor polynomials. But many polynomials do not factor over the real numbers; for instance, x 2 + 1 does not factor into a product of two linear polynomials with real coefficients; instead it requires complex numbers x 2 + 1 = ( x - i )( x + i ) . Consequently in this chapter we shall use complex numbers for our scalars, including entries in vectors and matrices. That is, we shift from studying vector spaces over the real numbers to vector spaces over the complex numbers. Any real number is a complex number and in this chapter most of the examples use only real numbers but nonetheless, the critical theorems require that the scalars be complex.
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Factoring and Complex Numbers
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Division Theorem for Polynomials Consider a polynomial p ( x ) = c n x n + · · · + c 1 x + c 0 with leading coefficient c n 6 = 0 . The degree of the polynomial is n . If n = 0 then p is a constant polynomial p ( x ) = c 0 . Constant polynomials that are not the zero polynomial, c 0 6 = 0 , have degree zero. We define the zero polynomial to have degree - .
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Division Theorem for Polynomials Consider a polynomial p ( x ) = c n x n + · · · + c 1 x + c 0 with leading coefficient c n 6 = 0 . The degree of the polynomial is n . If n = 0 then p is a constant polynomial p ( x ) = c 0 . Constant polynomials that are not the zero polynomial, c 0 6 = 0 , have degree zero. We define the zero polynomial to have degree - . Just as integers have a division operation — e.g., ‘ 4 goes 5 times into 21 with remainder 1 ’ — so do polynomials.
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Division Theorem for Polynomials Consider a polynomial p ( x ) = c n x n + · · · + c 1 x + c 0 with leading coefficient c n 6 = 0 . The degree of the polynomial is n . If n = 0 then p is a constant polynomial p ( x ) = c 0 . Constant polynomials that are not the zero polynomial, c 0 6 = 0 , have degree zero. We define the zero polynomial to have degree - . Just as integers have a division operation — e.g., ‘ 4 goes 5 times into 21 with remainder 1 ’ — so do polynomials. Example 3x x 2 + x ) 3x 3 + 2x 2 - x + 4
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Division Theorem for Polynomials Consider a polynomial p ( x ) = c n x n + · · · + c 1 x + c 0 with leading coefficient c n 6 = 0 . The degree of the polynomial is n . If n = 0 then p is a constant polynomial p ( x ) = c 0 . Constant polynomials that are not the zero polynomial, c 0 6 = 0 , have degree zero. We define the zero polynomial to have degree - .
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