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Example 1 Determine
if each of the following sets of vectors are linearly independent or linearly
dependent.
(a)  and . [Solution]
(b) and . [Solution]
(c) ,
,
and . [Solution]
(d) ,
,
and . [Solution]
Solution
To answer the question here we’ll need to set up the
equation
for each part,
combine the left side into a single vector and the set all the components of
the vector equal to zero (since it must be the zero vector, 0).
At this point we’ve got a system of equations that we can solve. If we only get the trivial solution the
vectors will be linearly independent and if we get more than one solution the
vectors will be linearly dependent.
(a) and .
We’ll do this
one in detail and then do the remaining parts quicker. We’ll first set up the equation and get the
left side combined into a single vector.
Now, set each
of the components equal to zero to arrive at the following system of
equations.
Solving this
system gives to following solution (we’ll leave it to you to verify this),
The trivial
solution is the only solution and so these two vectors are linearly
independent.
[Return to Problems]
(b) and .
Here is the
vector equation we need to solve.
The system of
equations that we’ll need to solve is,
and the
solution to this system is,
We’ve got more
than the trivial solution (note however that the trivial solution IS still a
solution, there’s just more than that this time) and so these vectors are
linearly dependent.
[Return to Problems]
(c) ,
,
and .
The only
difference between this one and the previous two are the fact that we now
have three vectors out of . Here is the vector equation for this part.
The system of
equations to solve for this part is,
So, not much
solving to do this time. It is clear
that the only solution will be the trivial solution and so these vectors are
linearly independent.
[Return to Problems]
(d) ,
,
and .
Here is the
vector equation for this final part.
The system of
equations that we’ll need to solve here is,
The solution to
this system is,
We’ve got more
than just the trivial solution and so these vectors are linearly dependent.
[Return to Problems]
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