Linear independence

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In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. 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.

Contents

1 Definition 2 The projective space of linear dependences 3 Example I 4 Example II 5 Example III: (calculus required) 6 See also

Definition

Let v₁, v₂, ..., v_n be real vectors of the same length. We say that they are linearly dependent if there exist numbers a₁, a₂, ..., a_n, not all equal to zero, such that:

Note that the zero on the right is the zero vector, not the number zero.

If such numbers do not exist, then the vectors are said to be linearly independent. This condition can be reformulated as follows: Whenever a₁, a₂, ..., a_n are numbers such that

we have a_i = 0 for i = 1, 2, ..., n.

More generally, let V be a vector space over a field K, and let {v_i}_i∈I be a family of elements of V. The family is linearly dependent over K if there exists a family {a_j}_j∈J of nonzero elements of K such that

where the index set J is a nonempty, finite subset of I.

A set X of elements of V is linearly independent if the corresponding family {x}_x∈X is linearly independent.

The concept of linear independence is important because a set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space.

The projective space of linear dependences

A linear dependence among vectors v₁, ..., v_n is a tuple (a₁, ..., a_n) with n scalar components, not all zero, such that

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, ...., v_n is a projective space.

Example I

The vectors (1, 1) and (−3, 2) in R² are linearly independent.

Proof:

Let a, b be two real numbers such that:

Then:

and

and

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,...,e_n are linearly independent.

Proof:

Suppose that a₁, a₂,...,a_n are elements of Rⁿ 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

                (1)

for all values of t. We need to show that a = 0 and b = 0. In order to do this, we differentiate both sides of (1) to get

              (2)

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.

See also

• orthogonalityhe:תלות לינארית

nl:lineair onafhankelijk pl:wektory liniowo niezależne

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