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Mathematik-Online problems:

Problem 17: Linear Independence

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#./aufgabe17_en.tex#Check if the following vectors are linearly independent. Determine the dimension of the spanned vector spaces and find a basis.
a)
$ (1,1,0)^{{\operatorname t}}$, $ (1,0,1)^{{\operatorname t}}$, $ (0,1,1)^{{\operatorname t}}$
b)
$ (1,2,3)^{{\operatorname t}}$, $ (2,3,4)^{{\operatorname t}}$, $ (3,4,5)^{{\operatorname t}}$
c)
$ (5,0,5,-4)^{{\operatorname t}}$, $ (0,5,-5,-3)^{{\operatorname t}}$, $ (5,-5,10,-1)^{{\operatorname t}}$, $ (-4,-3,-1,5)^{{\operatorname t}}$
Which $ \lambda\in\mathbb{R}$ cause linearly dependent vectors $ (2,\lambda,3)^{{\operatorname t}}$, $ (1,-1,2)^{{\operatorname t}}$ and $ (-\lambda,4,-3)^{{\operatorname t}}$. Describe the last vector as an linear combination of the first and the second vector, for these $ \lambda$.
(Authors: Höllig/Apprich/Höfert)

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[Solutions]
  automatisch erstellt am 12.  3. 2018