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Definition Let V be a vector space over a field K. If v_{1},v_{2},..,v_{n} are elements of V, we say that they are linearly dependent over K if there exist elements a_{1},a_{2},..,a_{n} in K not all equal to zero such that:
or, more concisely:
(Note that the zero on the right is the zero element in V, not the zero element in K.)
If there do not exist such field elements, then we say that v_{1},v_{2},...,v_{n} are linearly independent. An infinite subset of V is said to linearly independent if all its finite subsets are.
To focus the definition on linear independence, we can say that the vectors v_{1},v_{2},..,v_{n} are linearly independent, if and only if the following condition is satisfied:
Example I Show that the vectors (1,1) and (3,2) in R^{2} are linearly independent.
Proof:
Let a, b be two real numbers such that:
Then:
Solving for a and b, we find that a = 0 and b = 0.
Example II Let V=R^{n} and consider the following elements in V:
Then e_{1},e_{2},...,e_{n} are linearly independent.
Proof:
Suppose that a_{1}, a_{2}, ,a_{n} are elements of R^{n} such that
Since
then a_{i} = 0 for all i in {1,..,n}.
Example III: (Calculus required) Let V be the vector space of all functions of a real variable t. Then the functions e^{t} and e^{2t} in V are linearly independent.
Proof:
Suppose a and b are two real numbers such that
for all values of t. We need to show that a=0 and b=0. In order to do this,
we differentiate equation (1) to get
which also holds for all values of t.
Subtracting the first relation from the second relation, we obtain:
and, by plugging in t = 0, we get b = 0.
From the first relation we then get:
and again for t = 0 we find a = 0.
A linear dependence among vectors v_{1},...,v_{n} is a vector (a_{1},...,a_{n}) with n scalar components, not all zero, such that a_{1}v_{1}+...+a_{n}v_{n}=0. If such a linear dependence exists, then the n vectors are linearly dependent. It makes sense to identify two linear dependences if one arises as a nonzero multiple of the other, because in this case the two describe the same linear relationship among the vectors. Under this identification, the set of all linear dependences among v_{1}, ...., v_{n} is a projective space.
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