## 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

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