LINEAR DEPENDENCE, INDEPENDENCE
LECTURE-07
LINEAR DEPENDENCE, INDEPENDENCE
Linearly
Dependent:
A set of vectors is
1 ,
2,…,
n
to be linearly dependent if there exists a non
trivial linear combination of u1, u2, …., un
1,
2….,
that equals the zero vector.
Example: Prove that the
vectors
,
1, 1, 0
and
are linearly dependent.
Proof:
To prove linear dependence we must try to
find scalar
Such
that,
or,
Therefore,
and
This
can happen iff
, any non zero value for
. Say 1 will do.
Thus,
(Proved)
Definition:
If
1,
2, …..
n are
vectors of a
vector space
, then the linear combination,
1
1+α2
2+….+αn
n is called the scalars,
1,
2, ….,
n are zero. Otherwise, the linear
combination is said to be non trivial.
Line Through:
Given a vector
, the set of all scalar multiples of
is called the
line through
. Geometrically, in the case of
,
2and
3 it
is nothing but the straight line through the origin and
.
Collinear:
Two
vectors
1 and
2 are collinear if one of them lies in
the line through the other. Clearly,
is collinear
with any non-zero vector
.
Plane Through: Given
.
Posts
No comments