32 from textbook
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.
#32 from textbook
![b MATLAB: An Introduction with A X
Thomas W. Hungerford - Abstrac ×
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25. Let L be the ring of all matrices in M(Z) of the form
Show that the
function f:
not an isomorphism.
Z given by
= a is a surjective homomorphism but
26. Show that the homomorphism g in Example 7 is injective but not surjective.
27. (a) If g:
Sand f.S →T'are homomorphisms, show that fo g:
Tis a
homomorphism.
(b) If fand g are isomorphisms, show that fo g is also an isomorphism.
28. (a) Give an example of a homomorphism f:R→S such that R has an identity
but S does not. Does this contradict part (4) of Theorem 3.10?
(b) Give an example of a homomorphism f:R→S such that S has an identity
but Rdoes not.
S be an isomorphism of rings and let g:
R be the inverse
29. Let f:
function of f (as defined in Appendix B). Show that g is also an isomorphism.
[Hint: To show g(a + b) = g(a) + g(b), consider the images of the left- and
right-hand side under fand use the facts that fis a homomorphism and f•g is
the identity map]
30. Let f:R→S be a homomorphism of rings and let K = {rɛR|f(r) = 0s}.
Prove that K is a subring of R.
31. Let f:R→S be a homomorphism of rings and Ta subring of S.
Let P = {rɛR|S(r)ET}. Prove that P is a subring of R.
32. Assume n =1 (mod m). Show that the function f:Z,→ Zm given by
(xL) = [nx], is an injective homomorphism but not an isomorphism when
n2 2 (notation as in Exercise 12(e)).
33. (a) Let T be the ring of functions from R to R, as in Example 8 of Section 3.1.
R be the function defined by 0( f) =f(5). Prove that 0 is a
Let 0:
surjective homomorphism. Is 0 an isomorphism?
(b) Is part (a) true if 5 is replaced by any constant ceR?
34. If f: →Sis an isomomorphism of rings, which of the following properties
are preserved by this isomorphism? Justify your answers.
(a) aERis a zero divisor.
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3.3 Isomorphisms and Homomorphisms
83
(b) aER is idempotent.*
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