2,2 e set of all 2 × 2 matrices of the form Describing the Additive Inverse In E

Question

2,2 e set of all 2 × 2 matrices of the form Describing the Additive Inverse In Exercises 7-12, describe the additive inverse of a vector in the vector Space. 7. R4 10. M14 9. M23 12. M with the standard operations 2.2 27. The set of all 3 x 3 matrices of the form Testing for a Vector Space In Exercises 13-34, determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify at least one of the ten vector space c 0 d axioms that fails. 13. M46 with the standard operations 14. ML, with the standard operations with the standard operations 28. The set of all 3 x 3 matrices of the form 15. The set of all third-degree polynomials with the he set of all fifth-degree polynomials with the standard 17. The set of all first-degree polynomial functions ax, standard operations with the standard operations 29. The set of all 2 × 2 singular matrices with the standard a t 0, whose graphs pass through the origin with the standard operations 30 The set of all 2 x 2 nonsingular matrices with the The set of all first-degree polynomial functions ax + b, standard operations a, b 0, whose graphs do not pass through the origin 31. The set of all 2 x 2 diagonal matrices with the standard with the standard operations the standard operations through the origin with the standard operations 19. The set of all polynomials of degree four or less with 32. he set of all 3 x 3 upper triangular matrices with the tandard operations 20. The set of all quadratic functions whose graphs pass 33. C[O. 11, the set of all continuous functions defined on the interval [0, 1], with the standard operations 34·ct-1, 1], the set of all continuous functions defined on the interval [-1, 1], with the standard operations 21·The set (x, y): x 2 0, y is a real number

Explanation / Answer

16. The set of all 5th degree polynomials is NOT a vector space as the additive identity 0 being not a 5th degree polynomial does not belong to this set.

18. Let P denote the set of all 1st degree polynomials of the form p(x) = ax+b, where a,b0, whose graphs do not pass through the origin. Let p(x) = ax+b and q(x) = cx+d, where a,b,c,d 0 be 2 arbitrary elements of P. We may observe that the graphs of neither p(x), nor q(x) passes through the origin. Now, p(x)+q(x) = (a+c)x +(b+d) and when c = -a, and d = -b, both a+c and b+d equal 0 so that the graph of p(x)+q(x) passes through the origin. In such circumstances, p(x)+q(x) P so that P is not closed under vector addition. Hemce P is not a vector space.

30. Let A =

1

2

2

3

and B =

-1

-2

-2

-3

Then det(A) 0 and det(B) 0 so that both A and B belong to the set M of all non-singular 2x2 matrices. However, A+B =

-0

0

0

0

Apparently A+B is a singular matrix. Thus the set M is not closed under vector addition, and therefore, it is not a vector space.

32. The sum of two 3x3 upper triangular matrices is a 3x3 upper triangular matrix. Also the scalar multiple of a 3x3 upper triangular matrix is a 3x3 upper triangular matrix. Further the 3x3 zero matrix is also upper triangular. Hence the set of all 3x3 upper triangular matrices is a vector space.

1

2

2

3

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