Linear independence

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. Equivalently, a family is dependent if a member is in the linear span of the rest of the family, es decir,, a member is a linear combination of the rest of the family. A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space. Geometric meaning A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain place might say, "It is 5 miles north and 6 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered a 2-dimensional vector space (ignoring altitude). The person might add, "The place is 7.81 miles northeast of here." Although this last statement is true, it is not necessary. In this example the "5 miles north" vector and the "6 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, y viceversa. The third "7.81 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, Es decir, one of the three vectors is unnecessary. Note that in this example, any of the three vectors may be described as a linear combination of the other two. While it might be inconvenient, one could describe "6 miles east" in terms of north and northeast. (Por ejemplo, "Go 5 miles south (mathematically, −5 miles north) and then go 7.81 miles northeast.") Semejantemente, the north vector is a linear combination of the east and northeast vectors. Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. En general, n linearly independent vectors are required to describe a location in n-dimensional space. Example I The vectors (1, 1) y (−3, 2) in R2 are linearly independent. Proof Let λ1 and λ2 be two real numbers such that Taking each coordinate alone, this means Solving for λ1 and λ2, we find that λ1 = 0 and λ2 = 0. Alternative method using determinants An alternative method uses the fact that n vectors in Rn are linearly dependent if and only if the determinant of the matrix formed by taking the vectors as its columns is zero. En este caso, the matrix formed by the vectors is We may write a linear combination of the columns as We are interested in whether AΛ = 0 for some nonzero vector Λ. This depends on the determinant of A, which is Since the determinant is non-zero, the vectors (1, 1) y (−3, 2) are linearly independent. When the number of vectors equals the dimension of the vectors, the matrix is square and hence the determinant is defined. De otra manera, suppose we have m vectors of n coordinates, with m < n. Then A is an n×m matrix and Λ is a column vector with m entries, and we are again interested in AΛ = 0. As we saw previously, this is equivalent to a list of n equations. Consider the first m rows of A, the first m equations; any solution of the full list of equations must also be true of the reduced list. In fact, if 〈i1,…,im〉 is any list of m rows, then the equation must be true for those rows. Furthermore, the reverse is true. That is, we can test whether the m vectors are linearly dependent by testing whether for all possible lists of m rows. (In case m = n, this requires only one determinant, as above. If m > n, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available. Example II Let V = Rn and consider the following elements in V: Then e1, e2, ..., en are linearly independent. Proof Suppose that a1, a2, ..., an are elements of R such that Since then ai = 0 for all i in {1, ..., n}. Example III 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. Proof Suppose a and b are two real numbers such that aet + be2t = 0 for all values of t. We need to show that a = 0 and b = 0. In order to do this, we divide through by et (which is never zero) and subtract to obtain bet = −a In other words, the function bet must be independent of t, which only occurs when b = 0. It follows that a is also zero. The projective space of linear dependences A linear dependence among vectors v1, ..., vn is a tuple (a1, ..., an) 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, ...., vn is a projective space. See also orthogonality matroid (generalization of the concept) Wronskian de:Lineare Unabhängigkeit es:Dependencia e independencia lineal eo:Lineara sendependeco fr:Indépendance linéaire ko:일차 독립 he:תלות לינארית hu:Lineáris függetlenség nl:Lineaire onafhankelijkheid pt:Independência linear sl:Linearna neodvisnost fi:Lineaarinen riippumattomuus sv:Linjärt oberoende vi:Độc lập tuyến tính ur:لکیری آزادی zh:線性相關性 This page uses Creative Commons Licensed content from Wikipedia (ver autores).

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