4.2: Vector Algebra

Outcomes

1. Understand vector addition and scalar multiplication, algebraically.
2. Introduce the notion of linear combination of vectors.

Addition and scalar multiplication are two important algebraic operations done with vectors. Notice that these operations apply to vectors in $\mathbb{R}^{n}$, for any value of $n$. We will explore these operations in more detail in the following sections.

Addition of Vectors in $\mathbb{R}^n$

Addition of vectors in $\mathbb{R}^n$ is defined as follows.

Definition $\PageIndex{1}$: Addition of Vectors in $\mathbb{R}^n$

If $\vec{u}=\left [ \begin{array}{c} u_{1} \\ \vdots \\ u_{n} \end{array} \right ],\; \vec{v}= \left [ \begin{array}{c} v_{1} \\ \vdots \\ v_{n} \end{array} \right ] \in \mathbb{R}^{n}$ then $\vec{u}+\vec{v}\in \mathbb{R}^{n}$ and is defined by

$$\begin{aligned} \vec{u}+\vec{v} &= \left [ \begin{array}{c} u_{1} \\ \vdots \\ u_{n} \end{array} \right ] + \left [ \begin{array}{c} v_{1} \\ \vdots \\ v_{n} \end{array} \right ]\\ & = \left [ \begin{array}{c} u_{1}+v_{1} \\ \vdots \\ u_{n}+v_{n} \end{array} \right ]\end{aligned}$$

To add vectors, we simply add corresponding components. Therefore, in order to add vectors, they must be the same size.

Below is a video on finding the sum of two vectors as a linear combination of other vectors.

Addition of vectors satisfies some important properties which are outlined in the following theorem.

Theorem $\PageIndex{1}$: Properties of Vector Addition

The following properties hold for vectors $\vec{u},\vec{v}, \vec{w} \in \mathbb{R}^{n}$.

• The Commutative Law of Addition $$\vec{u}+\vec{v}=\vec{v}+\vec{u}\nonumber$$
• The Associative Law of Addition $$\left( \vec{u}+\vec{v}\right) +\vec{w}=\vec{u}+\left( \vec{v}+\vec{w}\right)\nonumber$$
• The Existence of an Additive Identity $$\vec{u}+\vec{0}=\vec{u} \label{vectoridentity}$$
• The Existence of an Additive Inverse $$\vec{u}+\left( -\vec{u}\right) =\vec{0}\nonumber$$

The additive identity shown in Equation $\eqref{vectoridentity}$ is also called the zero vector, the $n \times 1$ vector in which all components are equal to $0$. Further, $-\vec{u}$ is simply the vector with all components having same value as those of $\vec{u}$ but opposite sign; this is just $(-1)\vec{u}$. This will be made more explicit in the next section when we explore scalar multiplication of vectors. Note that subtraction is defined as $\vec{u}-\vec{v} = \vec{u}+\left( -\vec{v} \right)$.

Scalar Multiplication of Vectors in $\mathbb{R}^n$

Scalar multiplication of vectors in $\mathbb{R}^n$ is defined as follows.

Definition $\PageIndex{2}$: Scalar Multiplication of Vectors in $\mathbb{R}^n$

If $\vec{u}\in \mathbb{R}^{n}$ and $k\in \mathbb{R}$ is a scalar, then $k\vec{u}\in \mathbb{R}^{n}$ is defined by $$k\vec{u}=k\left [ \begin{array}{c} u_{1} \\ \vdots \\ u_{n} \end{array} \right ] = \left [ \begin{array}{c} ku_{1} \\ \vdots \\ ku_{n} \end{array} \right ]\nonumber$$

Just as with addition, scalar multiplication of vectors satisfies several important properties. These are outlined in the following theorem.

Theorem $\PageIndex{2}$: Properties of Scalar Multiplication

The following properties hold for vectors $\vec{u},\vec{v}\in \mathbb{R}^{n}$ and $k,p$ scalars.

• The Distributive Law over Vector Addition $$k \left( \vec{u}+\vec{v}\right) = k\vec{u}+ k\vec{v}\nonumber$$
• The Distributive Law over Scalar Addition $$\left( k + p \right)\vec{u} = k \vec{u}+p \vec{u}\nonumber$$
• The Associative Law for Scalar Multiplication $$k \left( p \vec{u}\right) = \left(k p \right)\vec{u}\nonumber$$
• Rule for Multiplication by $1$ $$1\vec{u}=\vec{u}\nonumber$$

Proof

We will show the proof of: $$k \left( \vec{u}+\vec{v}\right) = k \vec{u}+ k \vec{v}\nonumber$$ Note that: $$\begin{array}{ll} k \left( \vec{u}+\vec{v}\right) & =k \left [ u_{1}+v_{1} \cdots u_{n}+v_{n}\right ]^T \\ & = \left [ k \left( u_{1}+v_{1}\right) \cdots k \left( u_{n}+v_{n}\right) \right ]^T \\ & = \left [ k u_{1}+ k v_{1} \cdots k u_{n}+ k v_{n}\right ]^T \\ & = \left [ k u_{1} \cdots k u_{n} \right ]^T + \left [ k v_{1} \cdots k v_{n} \right ]^T \\ & = k \vec{u}+k \vec{v} \\ \end{array}\nonumber$$

Below is a video on the dot product.

Below is a video on vector scalar multiplication.

We now present a useful notion you may have seen earlier combining vector addition and scalar multiplication

Definition $\PageIndex{3}$: Linear Combination

A vector $\vec{v}$ is said to be a linear combination of the vectors $\vec{u}_1,\cdots , \vec{u}_n$ if there exist scalars, $a_{1},\cdots ,a_{n}$ such that $$\vec{v} = a_1 \vec{u}_1 + \cdots + a_n \vec{u}_n\nonumber$$

For example, $$3 \left [ \begin{array}{r} -4 \\ 1 \\ 0 \end{array} \right ] + 2 \left [ \begin{array}{r} -3 \\ 0\\ 1 \end{array} \right ] = \left [ \begin{array}{r} -18 \\ 3 \\ 2 \end{array} \right ].\nonumber$$ Thus we can say that $$\vec{v}= \left [ \begin{array}{r} -18 \\ 3 \\ 2 \end{array} \right ]\nonumber$$ is a linear combination of the vectors $$\vec{u}_1 = \left [ \begin{array}{r} -4 \\ 1 \\ 0 \end{array} \right ] \mbox{ and } \vec{u}_2 = \left [ \begin{array}{r} -3 \\ 0\\ 1 \end{array} \right ]\nonumber$$