Independence in arithmetic vector spaces

Task number: 2520

Decide, whether the following set of vectors is independent in the arithmetic vector spaces \(\mathbb R^4, \mathbb Z_3^4\) and \(\mathbb Z_5^4\).

If not, find an expression of some vector as a linear combination of the others.

  • Variant 1

    \(X_1=\{(0{,}1,2{,}1)^T,(1{,}2,0{,}0)^T,(1{,}1,2{,}0)^T,(1{,}2,1{,}1)^T\}\).

  • Variant 2

    \(X_2=\{(1{,}1,2{,}0)^T,(1{,}2,1{,}1)^T,(0{,}1,2{,}1)^T,(1{,}1,0{,}0)^T\}\).

  • Variant 3

    \(X_3=\{(1{,}0,2{,}0)^T,(2{,}1,0{,}2)^T,(0{,}2,2{,}1)^T,(2{,}2,1{,}1)^T\}\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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