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Mathematics-Online course: Prepcourse Mathematics - Linear Algebra and Geometry - Vector spaces

Subspace

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A subset $ U$ of a $ K$-vector space $ V$ is called a vector subspace (or simply a subspace) of $ V$, if $ U$ itself, endowed with the addition and the scalar multiplication defined in $ V$, forms a vector space.

if $ u, v\in U$ and $ \lambda\in K$, then it immediatetly follows that $ u + v \in U $ and $ \lambda\cdot u \in U$.

(Authors: App/Burkhardt/Kimmerle)
Subspaces often arise by specifying additional properties. Consider the vector space of real functions

$\displaystyle f:\,\mathbb{R}\to\mathbb{R}\, . $

Then, for example, the even functions ( $ f(x) = f(-x)$ for all $ x\in\mathbb{R}$) form a subspace. Further examples and counter-examples are given in the following table:

Property Subspace?
uneven yes
bounded yes
monotone no
continuous yes
positive no
linear yes

(Authors: App/Burkhardt/Höllig)
For any vector $ d\ne 0$ in a $ K$-vector space each line passing through origin $ O$

$\displaystyle v = \lambda d,\quad \lambda\in K\, $

forms a subspace.

But a line not passing through origin $ O$ does not form a subspace. For example, point $ (1,0)$ lies on the line

$\displaystyle g: x=(1,0)^{\operatorname t}+\mu(0,1)^{\operatorname t}\,,\quad\mu\in\mathbb{R}, $

point $ (2,0)$, however, does not.

(Authors: App/Burkhardt/Höllig)
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  automatically generated 10/23/2009