TR3.5

Surjectivity of Functions

Exr0n 2021-10-02 Sat 10:58

1 In the context of Linear Algebra (Axler 3.20) #defintion surjective   def

A function \(T : V \to W\) is called surjective if its range equals \(W\).

1.1 #aka onto   aka

1.2 Properties

1.2.1 A non-surjective map can be made surjective by changing the output space. (intuitive, not in book)

1.2.2 A map to a larger dimensional space is not surjective (Axler3.24)

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

  1. Intuition

    Surjectivity means that every element in the output space is mapped to, and that makes this intuitively true: If the number of possible inputs (and by extension, possible outputs) is smaller dimension than the output space, how can every output be mapped to?