General independence

Task number: 2519

Let \(u,v,w\) be linear independent vectors in a vector space\(V\) over the field \(\mathbb R\). Decide, whether the following sets of vectors are linearly independent or not.

  • Variant 1

    \(\{u,u+v,u+w\}\).

  • Variant 2

    \(\{u+v,u-v,w\}\).

  • Variant 3

    \(\{u+v,u-v,u+w,u-w\}\).

  • Variant 4

    \(\{u+v,u+w,v+w\}\).

  • Variant 5

    \(\{u-v,u-w,v-w\}\).

  • Variant 6

    \(\{u-2v+w,3u+v-2w,7u+14v-13w\}\).

  • Variant 7

    \(\{u,2u,w\}\)

  • Variant 8

    \(\{u,v+w\}\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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