nautolian said:Homework Statement
Show that if ker T != 0 then T is not an isomorphism.
Homework Equations
The Attempt at a Solution
If Ker T != 0 that means that there are multiple solutions for which T=0 meaning it is not injective and hence not isomorphic? Is that correct? I don't think it is, or is there a better way of proving this? Thanks.
An isomorphism is a type of function that establishes a one-to-one, onto relationship between two mathematical objects, such as groups, rings, or vector spaces. This means that the function is both injective (every element in the first object is mapped to a unique element in the second object) and surjective (every element in the second object has a corresponding element in the first object).
No, if Ker T is not equal to 0, then the function T cannot be an isomorphism. This is because the kernel of an isomorphism must be the trivial subgroup (consisting only of the identity element) in order for the function to be both injective and surjective. If Ker T is not 0, then there are elements in the first object that are mapped to the identity element in the second object, violating the injective property.
An isomorphism is a type of function that is invertible, meaning that it has an inverse function that maps elements back to their original values. If Ker T is not equal to 0, then the function T is not one-to-one and therefore cannot have an inverse, making it not an isomorphism.
No, an isomorphism can only exist between two objects that have the same structure. This means that they have the same operations and obey the same rules. For example, an isomorphism between two groups means that they have the same group operation (such as addition or multiplication) and the same group axioms (such as associativity and identity).
This statement highlights the important relationship between the kernel and the properties of an isomorphism. The fact that Ker T must equal 0 for T to be an isomorphism shows that the injective and surjective properties are essential for a function to be an isomorphism. It also emphasizes the importance of the structure of the objects being mapped in order for an isomorphism to exist.