In linear algebra, a set of elements of a vector space is linearly independent if none of the vectorss in the set can be written as a linear combination of finitely many other vectors in the set. For instance, in three-dimensional Euclidean space R3, the three vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1) are linearly independent, while (2, -1, 1), (1, 0, 1) and (3, -1, 2) are not (since the third vector is the sum of the first two). Vectors which are not linearly independent are called linearly dependent.
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2 Example I 3 Example II 4 Example III: (Calculus required) |
Definition
Let V be a vector space over a field K.
If v1,v2,..,vn are elements of V,
we say that they are linearly dependent over K if there exist elements a1,a2,..,an in K not all equal to zero such that:
If there do not exist such field elements, then we say that v1,v2,...,vn are linearly independent. An infinite subset of V is said to linearly independent if all its finite subsets are linearly independent.
To focus the definition on linear independence, we can say that the vectors v1,v2,..,vn are linearly independent, if and only if the following condition is satisfied:
Proof:
Let a, b be two real numbers such that:
Proof:
Suppose that a1, a2, ,an are elements of Rn such that
Proof:
Suppose a and b are two real numbers such that
we differentiate equation (1) to get
Subtracting the first relation from the second relation, we obtain:
From the first relation we then get:
A linear dependence among vectors v1,...,vn is a vector (a1,...,an) with n scalar components, not all zero, such that a1v1+...+anvn=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 non-zero 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 v1, ...., vn is a projective space.
See also:
Example I
Show that the vectors (1,1) and (-3,2) in R2 are linearly independent.
Then:
Solving for a and b, we find that a = 0 and b = 0.Example II
Let V=Rn and consider the following elements in V:
Then e1,e2,...,en are linearly independent.
Since
then ai = 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 et and e2t in V are linearly independent.
for all values of t. We need to show that a=0 and b=0. In order to do this,
which also holds for all values of t.
and, by plugging in t = 0, we get b = 0.
and again for t = 0 we find a = 0.