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.

[Column Vector]

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

[Row Vector]

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

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, φ, in V, such that x + φ = 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 = φ.
6. $cx\in V$
7. $c(x+y)=cx+cy$
8. $(c+d)x=cx+dx$
9. $c(dx)=cdx$
10. 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.