## Finite vector space

### Task number: 2506

How many elements does arithmetical vector space \(\mathbb Z_5^4\) have?

How many elements are there in any smallest and largest proper subspace of \(\mathbb Z_5^4\)?

Write out all elements of the smallest subspace of \(\mathbb Z_5^4\), which contains the following vectors \((0{,}0,0{,}4)^T,(2{,}4,3{,}2)^T\) and \((1{,}2,4{,}3)^T\).

#### Resolution

\(|\mathbb Z_5^4|=625\).

Smallest proper subspace of \(\mathbb Z_5^4\) is \(\{(0{,}0,0{,}0)^T\}\) (as it is the smallest subspace ever and it is a subspace of every vector space).

The largest proper subspace is for example \(\{(a,b,c,0)^T, a,b,c\in \mathbb Z_5\}\) and is of cardinality 125.

The subspace \([(0{,}0,0{,}4)^T,(2{,}4,3{,}2)^T,(1{,}2,4{,}3)^T]\) has 25 elements, namely:

\(\{(0{,}0,0{,}0)^T,(0{,}0,0{,}1)^T,(0{,}0,0{,}2)^T,(0{,}0,0{,}3)^T,(0{,}0,0{,}4)^T,\\ (2{,}4,3{,}2)^T,(2{,}4,3{,}3)^T,(2{,}4,3{,}4)^T,(2{,}4,3{,}0)^T,(2{,}4,3{,}1)^T,\\ (4{,}3,1{,}4)^T,(4{,}3,1{,}0)^T,(4{,}3,1{,}1)^T,(4{,}3,1{,}2)^T,(4{,}3,1{,}3)^T,\\ (1{,}2,4{,}1)^T,(1{,}2,4{,}2)^T,(1{,}2,4{,}3)^T,(1{,}2,4{,}4)^T,(1{,}2,4{,}0)^T,\\ (3{,}1,2{,}3)^T,(3{,}1,2{,}4)^T,(3{,}1,2{,}0)^T,(3{,}1,2{,}1)^T,(3{,}1,2{,}2)^T\}\)