Engineering Analysis/Vector Spaces

Vectors and Scalars
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.


 * $$\begin{bmatrix}a \\ b\\ c\\ \vdots\end{bmatrix}$$


 * $$\begin{bmatrix}a & b & c &\cdots\end{bmatrix}$$

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
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: $$x, y \in V$$
 * 1) 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.
 * 2) The addition operation is commutative for all elements in V.
 * 3) The addition operation is associative for all elements in V.
 * 4) There is a unique neutral element, &phi;, in V, such that x + &phi; = x. This is also called a zero element.
 * 5) For every x in V, then there is a negative element -x in V such that -x + x = &phi;.
 * 6) $$cx \in V$$
 * 7) $$c(x + y) = cx + cy$$
 * 8) $$(c + d)x = cx + dx$$
 * 9) $$c(dx) = cdx$$
 * 10) 1 &times; 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.