Linear Transformation Theorem-1

Let T be a linear transformation from avector space U into V over the field F.Then T is non-singular iff T is one-one.

Let T be a non-singular transformation from U into V.

Let α₁,α₂∊U such that

T(α₁)=T(α₂)

⇨ T(α₁)-T(α₂)=0

⇨ T(α₁-α₂)=0 (∵ T is linear transformation )

⇨ α₁-α₂=0 (∵ T is non-singular )

⇨ α₁=α₂

∴ T(α)=T(α) ⇨ α=α

So T is one-one.

Conversally,

Let T be a one-one transformation.

Let α∊U such that T(α)=0

∴ α∊U, T(α)=0

⇨ T(α)=T(0) [∵ T(0)=0]

⇨ α=0 [T is one-one ]

∴ α∊U and T(α)=0 ⇨ α=0

So T is non-singular. Proved

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