A Concrete Approach to Abstract Algebra |
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Page 143
... mixture of P and Q. ( ii ) Every vector F can be gotten in only one way as a mixture of P and Q , ( iii ) Every mixture of P and Q lies in the plane . To illustrate these : P by itself does not form Linear Dependence and Vector Spaces 143.
... mixture of P and Q. ( ii ) Every vector F can be gotten in only one way as a mixture of P and Q , ( iii ) Every mixture of P and Q lies in the plane . To illustrate these : P by itself does not form Linear Dependence and Vector Spaces 143.
Page 168
... mixture of three vectors might sometimes produce a vector space of only 2 dimen- sions . It is desirable to have a ... mixture of kA , B , C is a mixture of A , B , C , and con- versely . That a mixture of kA , B , C is a mixture of A ...
... mixture of three vectors might sometimes produce a vector space of only 2 dimen- sions . It is desirable to have a ... mixture of kA , B , C is a mixture of A , B , C , and con- versely . That a mixture of kA , B , C is a mixture of A ...
Page 169
... mixture of U , V , W is a mixture of P , Q , R is obvious . But since P = U , Q = V + sU , R = W , it is also obvious that any mixture of P , Q , R is expressible as a mixture of U , V , W. The proof for any number of vectors follows ...
... mixture of U , V , W is a mixture of P , Q , R is obvious . But since P = U , Q = V + sU , R = W , it is also obvious that any mixture of P , Q , R is expressible as a mixture of U , V , W. The proof for any number of vectors follows ...
Contents
Introduction | 1 |
The Viewpoint of Abstract Algebra | 5 |
Arithmetics and Polynomials | 26 |
Copyright | |
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Abstract Algebra Accordingly answer apply arith arithmetic modulo ax² axiom 12 belongs blue system calculating machine chapter class containing complex numbers consider constant polynomial construction corresponds course cubic cubic equation defined definition dimensions over F displacement divided division elementary algebra elements of F example fact field axioms field F form a basis geometrical gives Hence inches East integers irreducible irreducible polynomial isomorphic label leave remainder linear expressions linearly independent mathematics means ments metic mixture modular arithmetics modulo x² multiplication natural numbers nomial obtain odd number operations ordinary arithmetic P₁ plane poly possible procedure proof properties prove quadratic question rational coefficients rational numbers real numbers residue classes result satisfies an equation screws sequence simpler solution square root standard form statement structure subtract Suppose symbol things tion vector space whole number write