Consider the set
are real
numbers, with the same rules for addition and multiplication as in
a. Show that
b. Show that
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Elements Of Modern Algebra
- Let R be the set of all matrices of the form [abba], where a and b are real numbers. Assume that R is a commutative ring with unity with respect to matrix addition and multiplication. Answer the following questions and give a reason for any negative answers. Is 12 an integral domain? Is R a field?arrow_forwardConsider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.arrow_forward28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.arrow_forward
- Let [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].arrow_forwardGiven that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.arrow_forwardLet S be the set of all 2X2 matrices of the form [x0x0], where x is a real number.Assume that S is a ring with respect to matrix addition and multiplication. Answer the following questions, and give a reason for any negative answers. Is S a commutative ring? Does S have a unity? If so, identify the unity. Is S an integral domain? Is S a field? [Type here][Type here]arrow_forward
- Rather than use the standard definitions of addition and scalar multiplication in R2, let these two operations be defined as shown below. (x1,y1)+(x2,y2)=(x1+x2,y1+y2)c(x,y)=(cx,y) (x1,y1)+(x2,y2)=(x1,0)c(x,y)=(cx,cy) (x1,y1)+(x2,y2)=(x1+x2,y1+y2)c(x,y)=(cx,cy) With each of these new definitions, is R2 a vector space? Justify your answers.arrow_forward15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forward
- 43. Let . a. Show that is a noncommutative subring of . b. Find the unity element, if it exists.arrow_forwardRather than use the standard definitions of addition and scalar multiplication in R3, let these two operations be defined as shown below. (a) (x1,y1,z1)+(x2,y2,z2)=(x1+x2,y1+y2,z1+z2) c(x,y,z)=(cx,cy,0) (b) (x1,y1,z1)+(x2,y2,z2)=(0,0,0) c(x,y,z)=(cx,cy,cz) (c) (x1,y1,z1)+(x2,y2,z2)=(x1+x2+1,y1+y2+1,z1+z2+1) c(x,y,z)=(cx,cy,cz) (d) (x1,y1,z1)+(x2,y2,z2)=(x1+x2+1,y1+y2+1,z1+z2+1) c(x,y,z)=(cx+c1,cy+c1,cz+c1) With each of these new definitions, is R3 a vector space? Justify your answers.arrow_forward
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