Injective Definition and 92 Threads

Homework Statement Give an example of a linear vector space V and a linear transformation L: V-> V that is 1.injective, but not surjective (or 2. vice versa) Homework Equations -If L:V-> V is a linear transformation of a finitedimensional vector space, then L is surjective, L is...

I was a little curious on if I did the converse of this biconditonal statement correctly. Thanks in advance! =)Proposition: Suppose f:G->H is a homomorphism. Then, f is injective if and only if K={e}. Proof: Conversely, suppose K={e}, and suppose f(g)=f(g’). Now, if f(g)=f(g’)=e, then it follows...

Homework Statement Let E ⊆ R and f : E → R a decreasing function for all x ∈ E. Prove that f is injective. The attempt at a solution I tried that f were not injective. Then, there exist x < y such that f(x) = f(y) -This contradicts f being a decreasing function. I think this is...

Homework Statement Prove that the set of algebraic numbers is countably infinite. Homework Equations If there exists a bijective map between N and a set A, N and A have the same cardinality The Attempt at a Solution Rather than coming up with a bijective map between S =the set of...

Given two functions: f:A --> B g:B --> C How to show that if the (g ° f) is injection, then f is injection? I tried this: We need to show that g(f(a)) = g(f(b)) ==> a = b holds true for all a, b in A. But there's nothing said about function g.

Homework Statement Let I:=[0,1], let f: I→ℝ defined by f(x):= x when x is rational and 1-x when x is irrational. Show that f is injective on I and that f(fx) =x for all x in I. Show that f is continuous only at the point x =1/2 **I think i addressed all of these questions but I am unsure...

Hi, My question is to show that the linear transformation T: M2x2(F) -> P2(F) defined by T (a b c d) = (a-d) | (b-d)x | (c-d)x2 is surjective but not injective. thanks in advance.

I was struck with the following question: Is there a linear map that's injective, but not surjective? I know full well the difference between the concepts, but I'll explain why I have this question. Given two finite spaces V and W and a transformation T: V→W represented by a matrix \textbf{A}...

Prove that the inverse of a continuous injective function f:A -> ℂ on a compact domain A ⊂ ℂ is also continuous. So basically because we're in ℂ, A is closed and bounded, and since f is continuous, the range of f is also bounded. Given a z ∈ A, I can pick some arbitrary δ>0 and because f is...

Let U be a non-empty open set in Rn, if f:U->Rm is a diffeomorphism onto its image, show that df(p) is injective for all p in U. How can I attack this problem?

Homework Statement a) Let g: A => B, and f: B => C. Prove that f is one-to-one if f o g is one-to-one. b) Let g: A => B, and f: B => C. Prove that f is onto if f o g is onto. Homework Equations a) Since f o g is onto, then (f o g)(a) = (f o g)(b) => a = b. b) Since f o g is onto, every element...

Homework Statement Let A, B be sets, C,D\subset A and f:A\longrightarrow B be a function between them. Then f(C\cap D)=f(C)\cap f(D) if and only if f is injective. Homework Equations Another proposition, that I have proven that for any function f(C\cap D)\subset f(C)\cap f(D), and the...

Hello, I am looking for an example of two input injective function, f(x1,x2), R x R ->R. I am very grateful if you can find one. Thanks

Hello, I have g(t) is a continuous and a differential function under 1 variable. let h(x,y)=g(x^2+y^2) suppose that g(t) is Injective (thus monotonous) What is the shape of the contour lines of the graph of h(x,y)? -I have a sense that we're talking about simple cycles but I...

Hi! Define an integral operator K: L^2 (0,1) -> L^2(0,1) by: Kx(t) = Integral[ (1+ts)exp(ts)x(s) ds from t=0 to t=1]. Why is "obvious" that K is a one-to-one operator? I know K is one to one if Kx(t) = 0 implies x(t) = 0 but I don't see why this is true. Can you please explain why?

who can help me? ı want to prove this If f : X → Y is injective and A is a subset of X, then f −1(f(A)) = A. but how can I do this :(

Field of modulo p equiv classes, how injective linear map --> surjectivity Homework Statement Let Fp be the field of modulo p equivalence classes on Z. Recall that |Fp| = p. Let L: Fpn-->Fpn be a linear map. Prove that L is injective if and only if L is surjective. Homework Equations...

Hi, How do I prove that this functions is injective? a.) f : x --> x³ + x x ∈ R f(a) = a³ + a, f(b) = b³ + b f(a) = f(b) => a³ + a = b³ + b => a³ = b³ => a = b therefore f is one-to-one

Hey guys, new to the forum but hoping you can help. How do you prove that vector spaces V and U have a linear injective map given V is finite dimensional. I got the linear part but cannot really figure out the injectivity part, although I am thinking that it has to do with the kernel...

Homework Statement The following function f is a function from R to R. Determine whether f is injective (one-to-one), surjective (onto), or both. Please give reasons. Homework Equations f(x) = (x+1)/(x+2) if x != -2 f(x) = 1 when x = 2 The Attempt at a Solution f'(x) = 1/(x+2)2...

Homework Statement Give an example of a map f:A\rightarrowB and a map g:B\rightarrowC where g of f is injective but g is not injective. Homework Equations The Attempt at a Solution I'm not really sure what they are asking for.

Homework Statement We regard each polynomial p(t) an element of R(t) as defining a function p:R\rightarrow R, x \rightarrow p(x) prove that g:R[t]\rightarrow R[t], p(t) \rightarrow \int_{0}^{t}p(x)dx defines an injective linear transformation. Homework Equations The...

Homework Statement T ∈ L(V,W). Thread title. Homework Equations The Attempt at a Solution Note that T* is the adjoint operator. But there's one thing that I need to get out of the way before I even start the proof. Now consider <Tv, w>=<v, T*w> w in W, v in V. Now when they say T...

I was just wondering how you know if linear transformations injective?

In layman terms otherwise I have trouble understanding Injective: A function where no element on the domain is many to one. Surjective: All the elements in the codomain have at least one element from the domain that maps to them. I'd like to keep it simple so I can play it back to...

Q1. Claim: Suppose f : Rn -> Rm is injective. Then m >= n. Is this true? Q2. Claim: Suppose f : Rn -> Rn is injective and f(X) = [f1(X) f2(X) ... fn(X)]T. Then each fk must be injective. Is this true? Q3. I assume the above claims are known results or have known counterexamples. Can...

I'm not sure how i would go about this problem... Let f: A-> B (which i know means... f is a function from A to B which also means... that A is the domain and B is the range or image) Let A0\subsetA and B0\subsetB a. show that A0\subsetf-1(f(A0)) and the equality hold if f is...

Homework Statement How do I check if my function is surjective? How do I check if my function is injective? Suppose my function is a predicate and hence characteristic function of some set. How do I determine such a set? Homework Equations Does anyone know to write "The function...

Homework Statement Consider f: R^{m+1} - {0} -> R^{(m+1)(m+2)/2}, (x^{0},...,x^{m}) -> (x^{i} x^{j}) i<j in lexicografical order a) prove that f is an immersion b) prove that f(a) = f(b) if and only if b=±a, so that f restricted to Sm factors through an injective map g from Pm. c) show g...

Hi all, Can anyone point to an explanation of why if ker(phi) = {0}, then phi is injective?

could anyone give me a proof of this statement: The module is injective iff it is a direct summand of an injective cogenerator.

[SOLVED] Show map is injective Homework Statement Going crazy over this. Let 1<p<2 and q>=2 be its conjugate exponent. I want to show that the map T: L^p(E) --> (L^q(E))*: x-->T(x) where <T(x),y> = \int_Ex(t)y(t)dt is injective. This amount to showing that if \int_Ex(t)y(t)dt=0 for all...

can u gave me some examples of injective function that is not surjective. Is f(x)=y a injective function that is not surjective?

Hello! I hope I've posted this in the correct section... I'm a 3rd year undergraduate and we're currently studying Vector Spaces (in QM) and I just don't understand what injective (one-to-one) and surjective (onto) mean? As a result I have no idea what an isomorphism is! I realize this is...

hello, I've been reading some proofs and in keep finding this same argument tyo prove that a linear map is injective viz, we suppose that t(a,c) = 0 and then we deduce that a,c = 0,0. is it the case that the only way a linear map could be non injective is if it took two elements to zero? i.e. t...

This is something I understood before, but for some reason I forgot it. How do you prove this inequality holds, if f is injective? A_0 \subset f^{-1}(f(A_0))

I have recently taken great interest in studying the properties of the function f(x) = x^x , and I was wondering: is there any way to prove whether f(x) = x^x is an injective (i.e. one-to-one) function? I realize that if I can prove that if the inverse of f(x) = x^x is also a function, then...

Hi, I have no problems with recognising a bijective function -> one-to-one mapping e.g. x^3 is bijective wheras x^2 is not. But how do you tell weather a function is injective or surjective? I was reading various "math" stuff on this but it has left me only puzzled. Can somebody explain...

Indicate whether each statement is always true, sometimes true, or always false. IF T: R^n --> R^m is a linear transformation and m > n, then T is 1-1 Not sure to how prove this..

I had this question on a test today. Prove that if a function f:X-->Y is injective, then f(X\setminus A) \subset Y\setminus f(A), \forall A \subset X. This is how I did it: If x_1 is in A, then y_1=f(x_1) is in f(A). Because the function is injective, we can pick (cut Y into pieces) f(A) and...

is a cubic function injective for all real numbers?

Bijections result when the function is surjective and injective. How do I find a bijection in N and the set of all odd numbers? f(x) = 2x+1 Do I have to prove that this is one-to-one and onto? Am I on the right track?