## Inclusion of vector spaces

Determine, whether the spaces $$U_i$$ and $$V_i$$ are in an inclusion. If so, find a basis of the larger one that extend a basis of the smaller one.

These subspaces of $$\mathbb Z_5^7$$ are defined as follows:

• #### Variant 1

$$U_1=\mathcal L((4{,}1,0{,}3,4{,}0,0)^T,(4{,}3,1{,}0,2{,}3,1)^T,(4{,}1,4{,}0,3{,}2,4)^T,\\\strut\qquad (2{,}4,1{,}4,4{,}3,1)^T,(0{,}4,3{,}2,2{,}4,3)^T)$$

$$V_1=\{(x_1,…,x_7)^T\in\mathbb Z_5^7: x_1+3x_2+ x_3+2x_4+3x_5+ x_6+2x_7=0,\\\strut\qquad 3x_1+4x_2+3x_3+ x_4+4x_5+2x_6+4x_7=0,\ 2x_1+ x_2+4x_3 +4x_5 +2x_7=0\}$$

• #### Variant 2

$$U_2=\mathcal L((1{,}2,4{,}2,3{,}1,2)^T,(2{,}3,4{,}1,2{,}1,3)^T,(3{,}4,1{,}1,4{,}1,4)^T,\\\strut\qquad (4{,}0,2{,}3,3{,}4,1)^T,(4{,}3,1{,}3,2{,}3,2)^T)$$

$$V_2=\{(x_1,…,x_7)^T\in\mathbb Z_5^7: x_1+2x_2+ x_3 + x_5+2x_6+3x_7=0,\\\strut\qquad 4x_1+2x_2+ x_3+3x_4+2x_5 + x_7=0,\ x_1+ x_2+3x_3 + x_6 =0\}$$