Linear Independence
Summary
This section offers a different take on the equation 
that
we studied in section 1.5: we focus on the columns of A, and ask what
relation must exist between them when a non-trivial solution of the homogeneous
system exists. If we think of the matrix A in terms of its column vectors
(call them 
), then 
Definition: a set of vectors 
is
linearly independent if and only if (1) has only the trivial
solution (that is, 
).
Translation: The columns of A are linearly independent 
has only the trivial solution.
Definition: if (1) has a non-trivial solution, then the set of
vectors is linearly dependent.
The obvious consequence, contained in theorem 7, is that one of the vectors of
S can be expressed as a linear combination of the others. From (1) we
deduce that one of the coefficient 
is non-zero: if 
, then we
can solve for as
It's pretty clear that if S contains the zero vector, then the set is linearly dependent, since we can set the coefficient of the zero vector to 1, set all the other coefficients to zero, and we have a non-trivial solution to the homogeneous equation (Theorem 8).
It should also be clear from our discussions of spans that if we have more vectors in the set than the size of the space in which the vectors reside then the set will be linearly dependent. For example, if you have four distinct vectors in three-space, then the set will be linearly dependent. This is the substance of theorem 8.
