Linear Transformation

Linear Transformation or Vector Space Homomorphism

• Definition

Let U(F) and V(F) be two vector spaces over the same field F. A linear transformation from U into V is a function of T from U into V.

• Condition of linear transformation 

• T(aα + bβ)  =  aT(α) + bT(β)

∀ α and β in U and for all a,b in F.

• Linear operator 

  let V(f) be a vector space. A linear operator on V is a function T from V into V.

• T(aα + bβ)  =  aT(α) + bT(β)

for all α, β in V and a, b in F.

Thus T is a linear operator on V if T is a linear transformation from V into V itself.

Some particular Transformation

• Zero Transformation 

  let U(F) and V(F) be two vector spaces

  T: U into V

  T(α)=0 (∀ α ∈ U)

  is a linear transformation from U into V.

  Denoted by Ô(zero)

• Identity Operator 

Let V(F) be a vector space. The function I from V into V such that

I(α) = α ∀ α∈V

• Negative of a linear transformation

Let U(F) and V(F) be two vector spaces. Let T is a linear transformation from U into V.

(-T)(α) = -[T(α)] ∀α∈U