## Finite vector space

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:

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