Algebra Arrayán - download pdf or read online

By Ximena Carreño Campos, Ximena Cruz Schmidt

ISBN-10: 9562401685

ISBN-13: 9789562401685

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Example text

4. Cases (b), (c), (e) span R 2 . 5. 6. They are the same. Indeed, since v1 = u1 + 2 u2 , v2 = u1 + u2 , every vector v ∈ V can be written as a linear combination v = c1 v1 + c2 v2 = (c1 + c2 ) u1 + (2 c1 + c2 ) u2 and hence belongs to U . Conversely, since u1 = − v1 + 2 v2 , u2 = v1 − v2 , every vector u ∈ U can be written as a linear combination u = c1 u1 + c2 u2 = (− c1 + c2 ) v1 + (2 c1 − c2 ) v2 , and hence belongs to U . 7. (a) Every symmetric matrix has the form 49 a b b c ! =a 1 0 0 0 !

For the right-hand side we have to update at most k entries at each stage, so we have less than (n − 1)k multiplications and (n − 1)k additions. So we can get by with less than total (n − 1)(2 k + 2 k 2 ) multiplications and (n − 1)(k + 2 k 2 ) additions. (g) The inverse of a0banded matrix is1not necessarily banded. 16. 1 )T . (d) Partial pivoting reduces the effect of round off errors and results in a significantly more accurate answer. 17. 1429. 18. 9, (d) partial pivoting works markedly better, especially for the value of x.

Multiplied by − 5 is (b) Their sum is − 5 x y − 5 cos y − 15 x y + cos y + 3 ! 0 (c) The zero element is the constant function 0 = . 8. This is the same as the space of functions F(R 2 , R 2 ). Explicitly: Commutativity of Addition: ! ! v1 (x, y) w1 (x, y) v1 (x, y) + w1 (x, y) w1 (x, y) + = = + v2 (x, y) w2 (x, y) v2 (x, y) + w2 (x, y) w2 (x, y) Associativity of Addition: ! # = = " u1 (x, y) + v1 (x, y) + w1 (x, y) u2 (x, y) + v2 (x, y) + w2 (x, y) u1 (x, y) u2 (x, y) ! Additive Identity: 0 = (0, 0) for all x, y, and !

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Algebra Arrayán by Ximena Carreño Campos, Ximena Cruz Schmidt


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