13Vectors (in the geometric space). Changing the coordinates of a vector at replacement of a basis and the origin of coordinates.
Let
two Cartesian systems of coordinates: “old”
and “new”
be given.
Express
the vectors of “new” basis and vector
by the vectors of “old” basis:
Theorem. The coordinates of an arbitrary point in “old” system of coordinates are connected with its coordinates in “new” system by
(*)
For
a transition matrix
.
14Transition between orthonormal systems of coordinates on plane. Right (left) oriented pair of vectors on plane.
Consider
two orthonormal systems of coordinates
and
.
We/have
,
.
Then
the transition matrix is
and if
,
then the “old” coordinates will be connected with “new” as
In
the considered case both systems of coordinates could be combined by
sequential fulfilling parallel transposition of “old” system on
vector
and turning on angle
around
.
Consider the following:
Here
after combining the vectors
and
it will be required glassy reflecting the vector
regarding to the line passing through combined vectors. And the
transition formulas are the following:
An ordered pair of non-collinear vectors and b on plane with combined beginnings is called right oriented if the shortest turning from to is visible making in anticlockwise way. Otherwise this pair of vectors is called left-oriented
15 Linear dependence of vectors in the geometric space (plane and line). Theorems on properties of linearly dependent vectors.
Two vectors that are parallel to the same line are called collinear. Three vectors that are parallel to the same plane are called coplanar. A zero vector is collinear to every vector. A zero vector is coplanar to every pair of vectors.
An
expression of the form
where
are
some numbers is called a linear
combination
of vectors
.
If all the numbers
are
equal to zero simultaneously (that is equal to the condition
),
then such a linear combination is called trivial.
If there is at least one of the numbers
that
is distinct of zero (i.e.
),
then such a linear combination is called non-trivial.
Vectors
are linearly
dependent
if there is a non-trivial linear combination of these vectors
such that
.
Vectors
are linearly
independent
if
implies a triviality of the linear combinatio
,i.e.
.
Theorem. 1) One vector is linearly dependent iff it is zero.
2) Two vectors are linearly dependent iff they are collinear.
3) Three vectors are linearly dependent iff they are coplanar.
Lemma. Vectors are linearly dependent iff one of them is a linear combination of the rest vectors. Properties of linearly independent vectors
1)One vector is linearly independent iff it is non-zero.2)Two vectors are linearly independent iff they are non-collinear.3)Three vectors are linearly independent iff they are non-coplanar.
Theorem.
If there is a subset of linearly dependent vectors of
,
then the vectors
are linearly dependent.
Corollary.
If there is at least one zero vector of vectors
then the vectors
are linearly dependent.