## Independence in arithmetic vector spaces

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

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