Let $E$ be an affine space attached to a $K$-vector space $T$. Consider $K$ as an affine space attached to the $K$-vector space $K$. Write $B:=\{u\in K^E\ |\ \text{"$u$ is a affine"}\}$. Then $B$ is a right $K$-subspace of the $K$-vector space $K^E$. It is claimed that $\text{dim}_K(B)=1+\text{dim}_K(T)$. I cannot see why this holds.
Let $a\in E$. For each $v\in\text{Hom}_K(T,K)$ and $\lambda\in K$, let $\phi_{v,\lambda}$ be the mapping of $E$ into $K$ sending $x$ to $v(x-a)+\lambda$. Then $\phi:\text{Hom}_K(T,K)\oplus K\rightarrow B$ is an isomorphism. This implies that $\text{dim}_K(B)=1+\text{dim}_K(\text{Hom}_K(T,K))$. But this is not the same as what was claimed. Have I made a mistake?