Linear independence , Span, Basis,dimension

1. K.Anitha M.Sc., M.Phil.,
G.Nagalakshmi M.Sc., M.Phil.,

2.  In the theory of vector spaces,
a set of vectors is said to be linearly
dependent if at least one of the vectors in the
set can be defined as a linear combination of
the others; if no vector in the set can be written
in this way, then the vectors are said to
be linearly independent.

3.  Let V = Rn and consider the following
elements in V, known as the standard
basis vectors:
 e1=(1,0,0….0), e2 =(0,1,0,…0) …..
 en=(0,0,0,….1)
 e1,e2,23,….en are linearly independent

4.  In linear algebra, the linear span of
a set S of vectors in a vector space is the
smallest linear subspace that contains the
set. It can be characterized either as
the intersection of all linear subspaces that
contain S, or as the set of linear
combinations of elements of S. The linear
span of a set of vectors is therefore a vector
space.

5.  The real vector space R3 has {(-1,0,0), (0,1,0),
(0,0,1)} as a spanning set. This particular
spanning set is also a basis. If (-1,0,0) were
replaced by (1,0,0), it would also form
the canonical basis of R3.

6.  Another spanning set for the same space is
given by {(1,2,3), (0,1,2), (−1,1/2,3), (1,1,1)},
but this set is not a basis, because it
is linearly dependent.
 The set {(1,0,0), (0,1,0), (1,1,0)} is not a
spanning set of R3; instead its span is the
space of all vectors in R3 whose last
component is zero.

7.  Theorem 1: The subspace spanned by a non-
empty subset S of a vector space V is the set
of all linear combinations of vectors in S.
 Theorem 2: Every spanning set S of a vector
space V must contain at least as many
elements as any linearly independent set of
vectors from V.
 Theorem 3: Let V be a finite-dimensional
vector space. Any set of vectors that
spans V can be reduced to a basis for V by
discarding vectors if necessary

8.  A basis of a vector space is any linearly
independent subset of it that spans the whole
vector space. In other words, each vector in
the vector space can be written exactly in one
way as a linear combination of
the basis vectors. The dimension of a vector
space is the number of vectors in any of
its bases.

9.  Let V be a vector space. A minimal set of
vectors in V that spans V is called
a basis for V.
 Equivalently, a basis for V is a set of vectors
that
 is linearly independent;
 Spans V.

10.  The number of vectors in a basis for V is
called the dimension of V, denoted by dim(V).
For example, the dimension of Rn is n. The
dimension of the vector space of polynomials
in x with real coefficients having degree at
most two is 3. A vector space that consists of
only the zero vector has dimension zero.