Image and Surjectivity Theorem

• Let $T:V\to W$ be a linear transformation
• Suppose $W$ is a finite dimensional vector space
• Then, the following are equivalent
  1. $T$ is Surjective
  2. $Image(T)=W$
  3. $dim(Image(T))=dim(W)$

Proof

Proving $1⟹2$

• Suppose $T$ is surjective
• By defn of $Image(T)$, we know $Image(T)\subset W$
• We now argue that $W\in Image(T)$
  • Pick arbitrary $w\in W$, by defn of surjective
  • There is $\overrightarrow{v}\in V$ s.t $T(\overrightarrow{v})=\overrightarrow{w}$
  • Therefore, $\overrightarrow{w}\in Image(T)$
  • Thus, $W\subset Image(T)$
  • Thus, $Image(T)=W$

Proving $2⟹3$

• Suppose $Image(T)=W$
• Thus, $dim(Image(t))=dim(W)$ (note that we need to be finite dimensional for this to be defined)

Proving $3⟹1$

• Suppose $dim(Image(T))=dim(W)$
• Show that $T$ is surjective
• By Subspace Finite Dimensional Vector Space Lemma, they are equal