Set of Row Matrices under Entrywise Addition forms Vector Space

Corollary to Set of Matrices under Entrywise Addition forms Vector Space

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $\map {\MM_\GF} {1, n}$ be set of all row matrices over a field $\GF$.

Let $\struct {\map {\MM_\GF} {1, n}, +}$ denote the abelian group formed from $\map {\MM_\GF} {1, n}$ under matrix entrywise addition.

Let $\struct {\map {\MM_\GF} {1, n}, +, \times}_\GF$ denote the unitary module over $\GF$ where $\times$ denotes the matrix scalar product.

Then $\struct {\map {\MM_\GF} {1, n}, +, \times}_\GF$ forms a vector space.

Proof

This is an instance of Set of Matrices under Entrywise Addition forms Vector Space with $m = 1$.

$\blacksquare$

Sources

• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE: Example $5$