One-One & Onto Linear Transformation

One-one transformation:-

Let T be a transformation from a vector space U into V then T is said to be one-one transformation if

α₁,α₂∊U and α₁≠α₂ ⇒ T(α₁)≠T(α₂)

In other word

α₁,α₂∊U and T(α₁)=T(α₂) ⇒ α₁=α₂

Onto transformation:-

A transformation T:U→V is said to be onto if β∊V ⇒ ∃, α∊V such that

T(α)=β 

Invertible linear transformation:-

Let U & V be a vector spaces over the field F. Let T be linear transformation from U into V such that T is one-one and onto, then the inverse transformation exist ans is denoted by T^-1 .Also

β∊V ⇒ ∃ α∊U such that T(α)=β

∵        T(α)=β

So we can write

T^-1(β)=α