TR3.5

Injectivity (math)

Exr0n 2021-10-02 Sat 10:58

1 In the context of Linear Algebra (Axler 3.15)

1.1 #definition injective   def

A function \(T : V \to W\) is called injective if \(Tu = Tv\) implies \(u = v\)

1.2 #aka one-to-one   aka

1.3 Properties

1.3.1 A map is injective iff it's null space equals \(\{0\}\)

1.3.2 A map to a smaller dimensional space is not injective (Axler3.23)

Suppose \(V\) and \(W\) are finite-dimensional vector spaces such that \(\text{dim }V > \text{dim }W\). Then no linear map from \(V\) to \(W\) is injective.

  1. Intuition

    That makes sense, because if the output space has a smaller dimension, then there should be two inputs that go to the same output somewhere. Otherwise all the inputs just don't "fit".

1.4 Intuition

\(Tu = Tv \implies u = v\) means that if the outputs are the same, then the inputs are the same, aka only one input goes to that one output. That's why it's called "one-to-one": only one input goes to that one output