Dimension and Surjectivity Theorem

Theorem

• Suppose $V,W$ are finite dimensional vector spaces
• If $dim(V)<dim(W)$
• Then, $T:V\to W$ is not surjective

Intuition

There is no surjective linear transformation $R^2\to R^{100}$

Proof

• The rank-nullity theorem says: $dim(V)=dim(Image(T))+dim(Ket(T))$
• $⟹dim(V)\ge dim(Image(T))$ By whole numbers
• $⟹dim(W)>dim(V)>dim(image(T))$
• $⟹dim(W)>dim(Image(T))$
• Therefore, $dim(W)\ne dim(Image(T))$
• And thus, $T$ is not surjecitive1