# Engineering Analysis/Vector Spaces

Before reading this chapter, students should know the terms vector, scalar, and matrix. These terms are discussed in Linear Algebra. |

## Vectors and Scalars

[edit | edit source]A **scalar** is a single number value, such as 3, 5, or 10. A **vector** is an ordered set of scalars.

A vector is typically described as a matrix with a row or column size of 1. A vector with a column size of 1 is a **row vector**, and a vector with a row size of 1 is a **column vector**.

[Column Vector]

[Row Vector]

A "common vector" is another name for a column vector, and this book will simply use the word "vector" to refer to a common vector.

## Vector Spaces

[edit | edit source]A vector space is a set of vectors and two operations (addition and multiplication, typically) that follow a number of specific rules. We will typically denote vector spaces with a capital-italic letter: *V*, for instance. A space *V* is a vector space if all the following requirements are met. We will be using x and y as being arbitrary vectors in *V*. We will also use c and d as arbitrary scalar values. There are 10 requirements in all:

Given:

- There is an operation called "Addition" (signified with a "+" sign) between two vectors, x + y, such that if both the operands are in
*V*, then the result is also in*V*. - The addition operation is commutative for all elements in
*V*. - The addition operation is associative for all elements in
*V*. - There is a unique
**neutral element**, φ, in*V*, such that x + φ = x. This is also called a**zero**element. - For every x in
*V*, then there is a negative element -x in*V*such that -x + x = φ. - 1 × x = x

Some of these rules may seem obvious, but that's only because they have been generally accepted, and have been taught to people since they were children.