Showing that ( ker f ) ∗ = V ∗ / i m f ∗ {\displaystyle (\ker f)^{*}=V^{*}/\mathrm {im} f^{*}} for a linear map f : V → W {\displaystyle f:V\rightarrow W} . (Leo algknt and Jesse had a discussion.)
Let ι : ker f → V {\displaystyle \iota :\ker f\rightarrow V} be the inclusion of ker f {\displaystyle \ker f} into V {\displaystyle V} , then V ∗ / ker ι ∗ ≅ ( ker f ) ∗ {\displaystyle V^{*}/\ker \iota ^{*}\cong (\ker f)^{*}} .
We show that ker ι ∗ = i m f ∗ {\displaystyle \ker \iota ^{*}=\mathrm {im} f^{*}} .
ker ι ∗ = { ϕ ∈ V ∗ | ι ∗ ( ϕ ) = 0 } = { ϕ ∈ V ∗ | ϕ ∘ ι = 0 } = { ϕ ∈ V ∗ | ϕ | ker f = 0 , ϕ = f ∗ ( α ) , α ∈ W ∗ } = { ϕ ∈ V ∗ | ϕ | ker f = 0 , ϕ = α ∘ f , α ∈ W ∗ } = { f ∗ ( α ) ∈ V ∗ | ( α ∘ f ) | ker f = 0 , α ∈ W ∗ } = i m f ∗ . {\displaystyle {\begin{aligned}\ker \iota ^{*}&=\{\phi \in V^{*}\;\;|\;\;\iota ^{*}(\phi )=0\}\\&=\{\phi \in V^{*}\;\;|\;\;\phi \circ \iota =0\}\\&=\{\phi \in V^{*}\;\;|\;\;\phi |_{\ker f}=0,\;\phi =f^{*}(\alpha ),\;\alpha \in W^{*}\}\\&=\{\phi \in V^{*}\;\;|\;\;\phi |_{\ker f}=0,\;\phi =\alpha \circ f,\;\alpha \in W^{*}\}\\&=\{f^{*}(\alpha )\in V^{*}\;\;|\;\;(\alpha \circ f)|_{\ker f}=0,\;\alpha \in W^{*}\}\\&=\mathrm {im} f^{*}.\end{aligned}}}
for a linear map 
. (Leo algknt and Jesse had a discussion.)
be the inclusion of 
into 
, then 
.
.
