Serlo: EN: Vector space of a linear map

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Here, we consider the vector space of linear mappings between two given vector spaces.

The vector space of linear maps

Forming the vector space ${\mathrm{Hom}}_K(V,W)$

Let $K$ be a field and $V,W$ two $K$-vector spaces. We want to investigate the set of all $K$-linear mappings from $V$ to $W$. We give this set ${\mathrm{Hom}}_K(V,W)$ (like "homeomorphism", which is another word for "linear map"). So:

We will now prove that this set is again a $K$-vector space, so it has some very convenient properties. More precisely, ${\mathrm{Hom}}_K(V,W)$ is a subspace of the set of functions from $V$ to $W$, called $\mathrm{Fun}(V,W)$.

The dimension of the vector space of linear mappingsVorlage:Anker

In the following, we calculate the dimension of the vector space ${\mathrm{Hom}}_K(V,W)$ of linear maps between two finite-dimensional vector spaces.

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