40. Let d be an integer that is not a perfect square. Show that Q(Vd) (a+ bVdla, be Qj is a subfield of C. Hint: See Exercise 39.]
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
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the ring R. [Exercise 31 shows that the center of M(R) is the subring of scalar
matrices.]
33. Prove Theorem 3.1.
34. Show that M(Z,) (all 2 x 2 matrices with entries in Z,) is a 16-element
noncommutative ring with identity.
35. Prove or disprove:
(a) If Rand Sare integral domains, then Rx S is an integral domain.
(b) If Rand Sare fields, then R X S is a field.
36. Let T be the ring in Example 8 and let f, g be given by
ro
f(x) =
lx - 2 ifx> 2
[2 – x ifx< 2
if x > 2.
if x<2
g(x) =
to
Show that f, geT and that fg = 07. Therefore Tis not an integral domain.
37. (a) If Ris a ring, show that the ring M(R) of all 2 x 2 matrices with entries in
R is a ring.
(b) If Rhas an identity, show that M(R) also has an identity.
38. If R is a ring and aER, let ArR = {rɛR|ar = 0R}. Prove that AR is a subring
of R. Ag is called the right annihilator of a. [For an example, see Exercise 16 in
which the ring Sis the right annihilator of the mawix A.]
39. Let Q(V2) = (r + sV2|r, seQ}. Show that Q(V2) is a subfield of R.
[Hint: To show that the solution of (r + sV2)x = 1 is actually in Q(V2),
multiply 1/(r + sV2) by (r – sV/2)/(r -sV2).]
40. Let d'be an integer that is not a perfect square. Show that Q(Vd)
{a + bVd[a,bEQ} is a subfield of C. [Hìnt: See Exercise 39.]
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58
Chapter 3 Rings
41. Let S be the ring in Exercise 11.
(a) Verify that each of these matrices is a right identity in S:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2Fd12d5d2a-946d-4919-b8f6-75738f3b6448%2Ff0bzqno_processed.png&w=3840&q=75)
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