1 Explain why every linear functional is either surjective o

1 Explain why every linear functional is either surjective or the zero map.

Solution

A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with

f(x) =y..

The inverse of a linear function is also linear function.

For example

f(x) = x + 2

if we find inverse of this linear function which is also linear.

y = x + 2

x = y-2 is also linear .

Hence Linear function is Surjective.

let a linear function y = 0 . In that case it is a function of zero map.

hence linear functional is either surjective or the zero map.