# Definition:Vector Addition/Vector Space

## Definition

Let $\struct {F, +_F, \times_F}$ be a field.

Let $\struct {G, +_G}$ be an abelian group.

Let $V := \struct {G, +_G, \circ}_R$ be the corresponding **vector space over $F$**.

The group operation $+_G$ on $V$ is known as **vector addition** on $V$.

### Vector Addition on Real Vector Space

The usual context for **vector addition** is on a **real vector space** or **complex vector space**:

Let $\mathbf u$ and $\mathbf v$ be vector quantities of the same physical property.

### Component Definition

Let $\mathbf u$ and $\mathbf v$ be represented by their components considered to be embedded in a real $n$-space:

\(\ds \mathbf u\) | \(=\) | \(\ds \tuple {u_1, u_2, \ldots, u_n}\) | ||||||||||||

\(\ds \mathbf v\) | \(=\) | \(\ds \tuple {v_1, v_2, \ldots, v_n}\) |

Then the **(vector) sum** of $\mathbf u$ and $\mathbf v$ is defined as:

- $\mathbf u + \mathbf v := \tuple {u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n}$

Note that the $+$ on the right hand side is conventional addition of numbers, while the $+$ on the left hand side takes on a different meaning.

The distinction is implied by which operands are involved.

### Triangle Law

Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that:

- $\mathbf u$ is represented by $\vec {AB}$
- $\mathbf v$ is represented by $\vec {BC}$

that is, so that the initial point of $\mathbf v$ is identified with the terminal point of $\mathbf u$.

Then their **(vector) sum** $\mathbf u + \mathbf v$ is represented by the arrow $\vec {AC}$.

## Sources

- 1961: I.M. Gel'fand:
*Lectures on Linear Algebra*(2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces - 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.2$. Vector Spaces - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem