v = [x
, . . . , x
The set of all n-coordinate vectors defined in the field F, for which the
operations (10) give vectors within the set itself, form the n-dimensional vector
Every subset of V
with respect to the operations (10) is
called vector subspace of V
(F). For instance, in the two-dimensional plane, the
points of a cartesian axis form a subspace of the plane. Similar, subspaces of the
plane are given by any straight line passing through the origin, and subspaces
of the 3D space are given by any plane passing through the origin.
m vectors v
, . . . , v
, are said to be linearly independent if there is no choice
of m coefficients a
, . . . , a
(the choice of all zeros is excluded) such that
+ . . . + a
= 0 .
In the 2D plane, two points on different cartesian axes are linearly indepen-
dent, as are any two points belonging to different straight lines passing through
the origin. Viceversa, points belonging to the same straight line passing through
the origin are always linearly dependent.
It can be shown that, in an n-dimensional space V
(F), every set of m n
vectors is linearly dependent. A set of n linearly independent vectors (if they
exist) is called a basis of V
(F), in the sense that any other vector ofV
can be obtained as a linear combination of the base vectors. For instance, the
vectors [1, 0, 0], [0, 1, 0], and [0, 0, 1] form a basis for the 3D space, but there are
infinitely many other bases.
Between any two vectors of the same vector space the operation of dot prod-
uct is defined, and it returns the scalar sum of the component-by-component
products. As a formula, the dot product is written as
v w =
By convention, with v we indicate a column vector, while v denotes its transpo-
sition into a row. Therefore, the operation (13) can be referred as a row-column
A matrix can be considered as a list of vectors, organized in a table where
each element of the list occupies (by convention) one column. A matrix having
n rows and m columns defined over the field F can be written as
. . . a
. . .
. . . a
The multiplication of a matrix A F
by a (column) vector v V
In this book, the square brackets are used to indicate vectors and matrices. This is also the
notation used in Octave. Moreover, the variables representing vectors or matrices are always
typed in bold font.
A set I is closed with respect to an operation on its elements if the result of the operation
is always an element of I.