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Mathematics-Online course: Linear Algebra - Basic Structures - Vector Spaces

Vector Space

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Let $ K$ be a field. An Abelian group $ (V,+)$ is called $ K$-vector space (vector space over $ K$) if a scalar multiplication ,,$ \cdot$`` is defined so that for all $ \lambda$, $ \lambda_1$, $ \lambda_2 \in K$ and for all $ v$, $ v_1$, $ v_2 \in V$ the following holds true:


$\displaystyle (\lambda_1+\lambda_2)\cdot v$ $\displaystyle =$ $\displaystyle \lambda_1\cdot v + \lambda_2\cdot v$  
$\displaystyle \lambda\cdot(v_1+v_2)$ $\displaystyle =$ $\displaystyle \lambda\cdot v_1 + \lambda\cdot v_2$  
$\displaystyle (\lambda_1\cdot\lambda_2)\cdot v$ $\displaystyle =$ $\displaystyle \lambda_1\cdot(\lambda_2\cdot v)$  
$\displaystyle 1\cdot v$ $\displaystyle =$ $\displaystyle v\,.$  

If $ K=\mathbb{R}$ or $ K=\mathbb{C}$, then one speaks of a real or complex vectorspace, resp.

Note, that the plus stands for addition in $ V$ and for addition in $ K$. The same stands for the multiplication.

(Authors: App/Burkhardt/Kimmerle)

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  automatically generated 4/21/2005