Injective, Surjective and Bijective

Injective (One-to-One)

A linear transformation $A:V\to W$ is said to be injective (or one-to-one) if:

$\forall u,v\in V,Au=Av\Rightarrow u=v$

In other words, if two distinct input vectors $u$ and $v$ produce the same output vector, then $A$ is not injective. Injective functions preserve distinctness.

Surjective (Onto)

A linear transformation $A:V\to W$ is said to be surjective (or onto) if:

$\forall w\in W,\exists v\in V,Av=w$

In other words, for every vector $w$ in the codomain $W$, there exists a vector $v$ in the domain $V$ such that $Av=w$. Surjective functions cover the entire codomain.

Bijective (One-to-One Correspondence)

A linear transformation $A:V\to W$ is said to be bijective (or a one-to-one correspondence) if it is both injective and surjective.

In other words, a bijective function is a perfect matching between the domain and codomain, where every input vector corresponds to a unique output vector, and every output vector has a corresponding input vector.

These definitions are crucial in understanding the properties of linear transformations and their applications in various fields!