Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 4.1 Scalar-Vector Multiplication, 4.2 Column Vector Addition

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Algebraic Properties of Vector Addition and Scalar Multiplication

Let $\vec{x}, \vec{y}, \vec{z} \in R^{n \times 1}$ and $a, b \in R$ . Then, all of the following are properties of 4.2 Column Vector Addition:

Commutativity of vector addition: $\vec{x} + \vec{y} = \vec{y} + \vec{x}$
Associativity of vector addition: $\vec{x} + (\vec{y} + \vec{z}) = (\vec{x} + \vec{y}) + \vec{z}$
Additive identity: $\vec{x} + \vec{0} = \vec{x} + \vec{0} + \vec{x}$
Additive inverse: $\vec{x} + - \vec{x} = - \vec{x} + \vec{x} = 0$ with $- \vec{x} = - 1 \vec{x}$
Distributivity over vector addition: $a \cdot (\vec{x} + \vec{y}) = a \cdot \vec{x} + a \cdot \vec{y}$
Distributivity over scalar addition: $(a + b) \vec{x} = a \cdot \vec{x} + b \cdot \vec{x}$
Associativity of scalar multiplication: $a \cdot (b \cdot \vec{x}) = (a \cdot b) \cdot \vec{x} = a \cdot b \cdot \vec{x}$
Multiplicative identity of scalar multiplication: $1 \cdot \vec{x} = \vec{x}$

Remark. If the dimensions of $\vec{0}$ were not specified, assume that it is the same as $\vec{x}$ .
$⧫$

Commutativity of Vector Addition

\begin{proof}
To prove these statements, let $\vec{x}, \vec{y}, \vec{z} \in R^{n \times 1}$ and $a, b \in R$ . Let's begin with commutativity. Consider:

\vec{x} + \vec{y} = [\begin{matrix} x_{1} + y_{1} \\ x_{2} + y_{2} \\ ⋮ \\ x_{n} + y_{n} \end{matrix}]

By definition, we know $x_{i}, y_{i} \in R$ for all $i \in {1, 2, \dots, n}$ . Thus, we know $x_{i} + y_{i} = y_{i} + x_{i}$ because addition of real numbers is commutative. Because this is true for every index value, we see:

\vec{x} + \vec{y} = [\begin{matrix} x_{1} + y_{1} \\ x_{2} + y_{2} \\ ⋮ \\ x_{n} + y_{n} \end{matrix}] = [\begin{matrix} y_{1} + x_{1} \\ y_{2} + x_{2} \\ ⋮ \\ y_{n} + x_{n} \end{matrix}] = \vec{y} + \vec{x}

\end{proof}

Associativity of Vector Addition

\begin{proof}
Consider:

\vec{x} + (\vec{y} + \vec{z}) = [\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}] + [\begin{matrix} y_{1} + z_{1} \\ y_{2} + z_{2} \\ ⋮ \\ y_{n} + z_{n} \end{matrix}] = [\begin{matrix} x_{1} + (y_{1} + z_{1}) \\ x_{2} + (y_{2} + z_{2}) \\ ⋮ \\ x_{n} + (y_{n} + z_{n}) \end{matrix}]

Again, we can focus on the scalar addition $x_{i} + (y_{i} + z_{i} \in R)$ for all $i \in {1, 2, \dots, n}$ . Thus, we know the addition of scalars is associative:

x_{i} + (y_{i} + z_{i}) = (x_{i} + y_{i}) + z_{i}

Because this is true for every index value, we see:

\vec{x} + (\vec{y} + \vec{z}) = [\begin{matrix} x_{1} + (y_{1} + z_{1}) \\ x_{2} + (y_{2} + z_{2}) \\ ⋮ \\ x_{n} + (y_{n} + z_{n}) \end{matrix}] = [\begin{matrix} (x_{1} + y_{1}) + z_{1} \\ (x_{2} + y_{2}) + z_{2} \\ ⋮ \\ (x_{n} + y_{n} + z_{n}) \end{matrix}] = (\vec{x} + \vec{y}) + \vec{z}

\end{proof}

Remark. If the dimensions of 0→ were not specified, assume that it is the same as x→. ⧫

Commutativity of Vector Addition

Associativity of Vector Addition

Additive Identity

Remark. If the dimensions of $\vec{0}$ were not specified, assume that it is the same as $\vec{x}$ .
$⧫$