Matrices over a field form a vector space

Dependencies:

1. Matrix
2. Vector Space

Let $F$ be a field. Then $\mathbb{M}_{m, n}(F)$ is a vector space on $F$.

Proof

• Addition is closed
• Addition is associative
• Addition is commutative
• Zero matrix exists
• Additive inverse exists

Therefore, $\mathbb{M}_{m, n}(F)$ is an abelian group.

• Scalar product is associative: $r(sA) = (rs)A$.
• Scalar product is distributive: $(r+s)A = rA + sA$ and $r(A+B) = rA + rB$.
• Unit scalar: $1A = A$.

Therefore, $\mathbb{M}_{m, n}(F)$ is a vector space.

Dependency for:

1. Basis of F^n
2. Matrices form an inner-product space
3. Row space

Info:

• Depth: 4
• Number of transitive dependencies: 6

Transitive dependencies:

1. Group
2. Ring
3. Field
4. Vector Space
5. Semiring
6. Matrix