The Vector Space of m x n Matrices

From the Vector Spaces page, recall the definition of a Vector Space:

Definition: A nonempty set $V$ is considered a vector space if the two operations: 1. addition of the objects $\mathbf{u}$ and $\mathbf{v}$ that produces the sum $\mathbf{u} + \mathbf{v}$, and, 2. multiplication of these objects $\mathbf{u}$ with a scalar $a$ that produces the product $a \mathbf{u}$, are both defined and the ten axioms below hold. Furthermore, if $V$ is a vector space then the objects in $V$ are called vectors: 1. $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$ (Commutativity of vector addition). 2. $\mathbf{u} + (\mathbf{v} + \mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w}$ (Associativity of vector addition). 3. There exists a zero vector $\mathbf{0}$ such that $\mathbf{0} + \mathbf{u} = \mathbf{u} + \mathbf{0} = \mathbf{u}$ (Existence of an additive identity). 4. For every $\mathbf{u} \in V$, there exists a vector $-\mathbf{u}$ such that $\mathbf{u} + (-\mathbf{u}) = (-\mathbf{u}) + \mathbf{u} = \mathbf{0}$ (Existence of an additive inverses). 5. $a(b\mathbf{u}) = (ab)\mathbf{u}$. (Associativity of scalar multiplication) 6. $1\mathbf{u} = \mathbf{u}$ (Existence of a multiplicative identity). 7. $a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$ (Distributivity of a scalar multiplication over vector addition). 8. $(a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}$. (Distributivity of scalar multiplication over field addition) 9. If $\mathbf{u}, \mathbf{v} \in V$, then $(\mathbf{u} + \mathbf{v}) \in V$ (Closure under addition). 10. If $a$ is any scalar and $\mathbf{u} \in V$, then $a\mathbf{u} \in V$ (Closure under scalar multiplication).

We are now going to verify that the set of all $m \times n$ matrices denoted $M_{mn}$ is a vector space. First let $u, v, w \in M_{mn}$, and let $a, b \in \mathbb{F}$. We will now begin to verify all ten axioms.

• 1.

(1)

\begin{align} u + v = \begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} + \begin{bmatrix} v_{11} & v_{12} & \cdots & v_{1n}\\ v_{21} & v_{22} & \cdots & v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ v_{m1} & v_{m2} & \cdots & v_{mn} \end{bmatrix} = \begin{bmatrix} u_{11} + v_{11} & u_{12} + v_{12} & \cdots &u_{1n} + v_{1n}\\ u_{21} + v_{21} & u_{22} +v_{22} & \cdots & u_{2n} + v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} + v_{m1} & u_{m2} +v_{m2} & \cdots & u_{mn} + v_{mn} \end{bmatrix} \\ = \begin{bmatrix} v_{11} + u_{11} & v_{12} + u_{12} & \cdots &v_{1n} + u_{1n}\\ v_{21} + u_{21} & v_{22} +u_{22} & \cdots & v_{2n} + u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ v_{m1} + u_{m1} & v_{m2} +u_{m2} & \cdots & v_{mn} + u_{mn} \end{bmatrix} = \begin{bmatrix} v_{11} & v_{12} & \cdots & v_{1n}\\ v_{21} & v_{22} & \cdots & v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ v_{m1} & v_{m2} & \cdots & v_{mn} \end{bmatrix} + \begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} = v + u \end{align}

• 2.

(2)

\begin{align} u + (v + w) = \begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} + \begin{bmatrix} v_{11} + w_{11} & v_{12} + w_{12} & \cdots &v_{1n} + w_{1n}\\ v_{21} + w_{21} & v_{22} +w_{22} & \cdots & v_{2n} + w_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ v_{m1} + w_{m1} & v_{m2} +w_{m2} & \cdots & v_{mn} + w_{mn} \end{bmatrix} \\ = \begin{bmatrix} u_{11} + [v_{11} + w_{11}] &u_{12} + [v_{12} + w_{12}] & \cdots & u_{1n} + [v_{1n} + w_{1n}]\\ u_{21} + [v_{21} + w_{21}] & u_{22} + [v_{22} +w_{22}] & \cdots & u_{2n} + [v_{2n} + w_{2n}]\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} + [v_{m1} + w_{m1}] & u_{m2} + [v_{m2} +w_{m2}] & \cdots & u_{mn} + [v_{mn} + w_{mn}] \end{bmatrix} = \begin{bmatrix} [u_{11} + v_{11}] + w_{11} & [u_{12} + v_{12}] + w_{12} & \cdots & [u_{1n} + v_{1n}] + w_{1n}\\ [u_{21} + v_{21}] + w_{21} & [u_{22} + v_{22}] +w_{22} & \cdots & [u_{2n} + v_{2n}] + w_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ [u_{m1} + v_{m1}] + w_{m1} & [u_{m2} + v_{m2}] +w_{m2} & \cdots & [u_{mn} + v_{mn}] + w_{mn} \end{bmatrix} \\ = \begin{bmatrix} u_{11} + v_{11} & u_{12} + v_{12} & \cdots &u_{1n} + v_{1n}\\ u_{21} + v_{21} & u_{22} +v_{22} & \cdots & u_{2n} + v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} + v_{m1} & u_{m2} +v_{m2} & \cdots & u_{mn} + v_{mn} \end{bmatrix} + \begin{bmatrix} w_{11} & w_{12} & \cdots & w_{1n}\\ w_{21} & w_{22} & \cdots & w_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ w_{m1} & w_{m2} & \cdots & w_{mn} \end{bmatrix} = (u + v) + w \end{align}

• 3. The zero vector is $0 = \begin{bmatrix} 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & 0 & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{bmatrix}$, and thus we have that $u + 0 = \begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} + \begin{bmatrix} 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & 0 & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{bmatrix} = \begin{bmatrix} u_{11} +0 & u_{12}+0 & \cdots & u_{1n}+0\\ u_{21}+0 & u_{22}+0 & \cdots & u_{2n}+0\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1}+0 & u_{m2}+0 & \cdots & u_{mn}+0 \end{bmatrix} = u$.

• 4. The additive inverse of $u$ is $-u = \begin{bmatrix} -u_{11} & -u_{12} & \cdots & -u_{1n}\\ -u_{21} & -u_{22} & \cdots & -u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ -u_{m1} & -u_{m2} & \cdots & -u_{mn} \end{bmatrix}$ and thus we have that $u + (-u) = \begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} + \begin{bmatrix} -u_{11} & -u_{12} & \cdots & -u_{1n}\\ -u_{21} & -u_{22} & \cdots & -u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ -u_{m1} & -u_{m2} & \cdots & -u_{mn} \end{bmatrix} = \begin{bmatrix} 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & 0 & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{bmatrix} = 0$.

• 5. $a(bu) = a\begin{bmatrix} bu_{11} & bu_{12} & \cdots & bu_{1n}\\ bu_{21} & bu_{22} & \cdots & bu_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ bu_{m1} & bu_{m2} & \cdots & bu_{mn} \end{bmatrix} = \begin{bmatrix} abu_{11} & abu_{12} & \cdots & abu_{1n}\\ abu_{21} & abu_{22} & \cdots & abu_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ abu_{m1} & abu_{m2} & \cdots & abu_{mn} \end{bmatrix} = (ab)\begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} = (ab)u$.

• 6. The multiplicative identity is the scalar $k = 1$, that is $1 \begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} = \begin{bmatrix} 1 \cdot u_{11} & 1 \cdot u_{12} & \cdots & 1 \cdot u_{1n}\\ 1 \cdot u_{21} & 1 \cdot u_{22} & \cdots & 1 \cdot u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ 1 \cdot u_{m1} & 1 \cdot u_{m2} & \cdots & 1 \cdot u_{mn} \end{bmatrix} = \begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} = u$.

• 7. $a(u + v) = a\begin{bmatrix} u_{11} + v_{11} & u_{12} + v_{12} & \cdots &u_{1n} + v_{1n}\\ u_{21} + v_{21} & u_{22} +v_{22} & \cdots & u_{2n} + v_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} + v_{m1} & u_{m2} +v_{m2} & \cdots & u_{mn} + v_{mn} \end{bmatrix} = \begin{bmatrix} au_{11} + av_{11} & au_{12} + av_{12} & \cdots &au_{1n} + av_{1n}\\ au_{21} + av_{21} & au_{22} +av_{22} & \cdots & au_{2n} + av_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ au_{m1} + av_{m1} & au_{m2} +av_{m2} & \cdots & au_{mn} + av_{mn} \end{bmatrix} = au + av$.

• 8.

(3)

\begin{align} (a + b)u = (a + b)\begin{bmatrix} u_{11} & u_{12} & \cdots & u_{1n}\\ u_{21} & u_{22} & \cdots & u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ u_{m1} & u_{m2} & \cdots & u_{mn} \end{bmatrix} = \begin{bmatrix} (a+b)u_{11} & (a+b)u_{12} & \cdots & (a+b)u_{1n}\\ (a+b)u_{21} & (a+b)u_{22} & \cdots & (a+b)u_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ (a+b)u_{m1} & (a+b)u_{m2} & \cdots & (a+b)u_{mn} \end{bmatrix}\\ = \begin{bmatrix} au_{11} bu_{11} & au_{12} +bu_{12} & \cdots & au_{1n} + bu_{1n}\\ au_{21} + bu_{21} & au_{22} + bu_{22} & \cdots & au_{2n} + bu_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ au_{m1} + bu_{m1} & au_{m2} + bu_{m2} & \cdots & au_{mn} + bu_{mn}\end{bmatrix} = au + bu \end{align}

• 9. $u + v$ forms an $m \times n$ matrix and so $(u + v) \in M_{mn}$.

• 10. $au$ forms an $m \times n$ matrix and so $(au) \in M_{mn}$.

Therefore we have verified that all 10 vector space axioms hold for the set of $m \times n$ matrices in $M_{mn}$ under the defined operations of addition and scalar multiplication, and so $M_{mn}$ is a vector space.