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4.3: Linear Independence in Vector Spaces 107
This is equivalent to the homogeneous linear system
-
f ci + c 2 + 2c 3 = 0
\ 2ci + 2c 2 - 4c 3 = 0
Since the number of unknowns is greater than the number
of equations, this system has a non-trivial solution by 2.1.4.
Hence the vectors are linearly dependent.
These examples suggest that the question of deciding
whether a set of vectors is linearly dependent is equivalent to
asking if a certain homogeneous linear system has non-trivial
solutions. Further evidence for this is provided by the proof
of the next result.
Theorem 4.3.1
n
Let Ai,A 2,... ,A m be vectors in the vector space F where
F is some field. Put A — [A\\A 2\... \A m], an n x m matrix.
Then Ai, A 2,..., A m are linearly dependent if and only if the
number of pivots of A in row echelon form is less than m.
Proof
Consider the equation C\A\ + c 2A 2 + • • • + c mA m = 0 where
c\, c 2, .. , c m are scalars. Equating entries of the vector on
.
the left side of the equation to zero, we find that this equation
is equivalent to the homogeneous linear system
/ c i \
A . = 0 .
\c J
m
By 2.1.3 the condition for this linear system to have a non-
trivial solution ci, c 2,..., c m is that the number of pivots be
less than m . Hence this is the condition for the set of column
vectors to be linearly dependent.
