1. ### I Proof about injectivity

I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...
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### B Surjective/injective operators

Hi, I found in Kreyszig that if for any ##x_1\ and\ x_2\ \in \mathscr{D}(T)## then an injective operator gives: ##x_1 \ne x_2 \rightarrow Tx_1 \ne Tx_2 ## and ##x_1 = x_2 \rightarrow Tx_1 = Tx_2 ## If one has an operator T, is there an inequality or equality one can deduce from this...
3. ### I Does this theorem need that Ker{F}=0?

I have encountered this theorem in Serge Lang's linear algebra: Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W. In the proof he starts with C1F(v1) +...
4. ### Injective & Surjective Functions

Just wondering if anyone could help me get in the right direction with these questions and/or point me to some material that will help me better understand how to approach these questions In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R. (a) Show that a function...
5. ### Proving local injectivity of curve

Homework Statement Let γ : I → Rn be a regular smooth curve. Show that the map γ is locally injective, that is for all t0 ∈ I there is some ε > 0 so that γ is injective when restricted to (t0 − ε , t0 + ε ) ∩ I. Homework Equations The Attempt at a Solution [/B] So I know a function (or a...
6. ### Show a functions inverse is injective iff f is surjective

Hello all, Can anyone give me a pointer on how to start this proof?: f:E\rightarrow F we consider f^{-1} as a function from P(F) to P(E). Show f^(-1) is injective iff f is surjective.