We have seen how vector equations relate to a system of equations and how to frame different questions in terms of whether it is possible to find a linear combination from a set
that equals a target vector. This was the same as asking if our target vector was in the span of
When we looked at vector equations, we also looked at whether there was a unique linear combination or whether there are many ways to write a target vector as a linear combination.
Definition 1.8.1.
A set of vectors
is
linearly independent if the only linear combination of elements of
that equals the zero vector is the trivial linear combination. In other words,
being a linear independent set implies that if
where
then all
A set of vectors
is
linearly dependent if the set is not linearly independent. More specifically, there exists a solution to
where
and at least one of the