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

Linear Hull

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Let $ V$ be a vector space.

The set of all linear combinations of vectors $ v_1, \ldots , v_m \in V$ is called the linear hull of the vectors $ v_i$; notation: $ \operatorname{span}(v_1,\ldots,v_n)$.

If $ M \subset V$, then the set of all linear combinations from $ M$ is called linear hull of $ M$.

Linear hulls are subspaces.

(Authors: App/Burkhardt/Kimmerle)
Consider two different lines through the origin: $ g_1$ and $ g_2$. For example:

$\displaystyle g_1:x=\lambda_1(1,0,0)^{\operatorname t},\, g_2: x=\lambda_2(0,1,0)^{\operatorname t},\quad \lambda\in\mathbb{R}. $

Thus, these lines are the linear hulls of vectors $ v_1$ and $ v_2$, resp. In our example:

$\displaystyle v_1=(1,0,0)^{\operatorname t},\, v_2=(0,1,0)^{\operatorname t}\,.$

Then, the linear hull of the two vectors $ v_1$ and $ v_2$ is the plane $ E$ containing the lines $ g_1$ and $ g_2$. In our example:

$\displaystyle E$ $\displaystyle :x=\lambda_1(1,0,0)^{\operatorname t}+\lambda_2(0,1,0)^{\operatorname t},\quad \lambda_i\in\mathbb{R}\, ,$    

or


$\displaystyle E$ $\displaystyle = \{x:\, x_3=0\}\,.$    

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005