How Do You Determine if a Linear Transformation is Injective?

  • Context: Undergrad 
  • Thread starter Thread starter mbud
  • Start date Start date
  • Tags Tags
    Injective Linear
Click For Summary

Discussion Overview

The discussion revolves around determining whether a linear transformation is injective. It includes theoretical aspects of linear transformations and their properties, particularly focusing on the kernel and its implications for injectivity.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant inquires about the criteria for injectivity in linear transformations.
  • Another participant suggests that examining the kernel is essential, noting that for a map from V to V, the determinant can indicate whether the kernel is zero, although it does not specify what the kernel is.
  • A further contribution clarifies that a function is injective if the condition f(x) = f(y) implies x = y holds true, and for linear functions, this translates to the kernel being trivial, specifically {0}.

Areas of Agreement / Disagreement

Participants present various perspectives on the injectivity of linear transformations, but there is no explicit consensus on the best approach or interpretation of the kernel's role.

Contextual Notes

The discussion does not resolve the implications of the determinant in relation to injectivity, nor does it clarify the conditions under which the kernel may be considered trivial.

mbud
Messages
6
Reaction score
0
I was just wondering how you know if linear transformations injective?
 
Physics news on Phys.org
You work out its kernel. If it's a map from V to V then you can work out its determinant which tells you if the kernel is zero or not (but not what it is).
 
Last edited:
thanks
 
A function, f, in general, is injective if f(x)= f(y) implies x= y. If f is linear, then f(x)= f(y) gives f(x)- f(y)= f(x-y)= 0 while x= y is the same as x- y= 0. That is why a linear function is injective if and only if its kernel is trivial: {0}, as matt grime said.
 

Similar threads

Undergrad DimKer(T) = 0 <--> T is injective
  • · Replies 6 ·
Replies
6
Views
5K
Undergrad Prove that if T is injective, T*T is invertible
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Proof of Invertibility: Linear Map's Surjectivity and Injectivity Condition
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Does this theorem need that Ker{F}=0?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Injective linear mapping - image
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Want to understand how to express the derivative as a matrix
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Wronskian: null space equals the span of independent functions
  • · Replies 23 ·
Replies
23
Views
2K
High School What unique contributions to math did linear algebra make?
  • · Replies 19 ·
Replies
19
Views
4K
High School How Can You Determine if an Operator is Surjective, Injective, or Bijective?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Injectivity equivalent to having a left inverse
  • · Replies 1 ·
Replies
1
Views
2K