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Linear Dependence.
A set of k vectors { v 1, v 2,.... v k} in Rn is linearly dependent if a set of scalars {s1, s2,.... sk} can be found, not all zero, such that:
s1 * v 1 + s2 * v 2 +.... + sk * v k = 0.
If no such set of scalars exists, the k vectors { v 1, v 2,.... v k} are linearly independent.
Linear Combination.
A vector v depends linearly on vectors { v 1, v 2,.... v k} if scalars {s1, s2,.... sk} exist such that:
v = s1 * v 1 + s2 * v 2 +.... + sk * v k.
D Example.
The vector u = (18.75, 16.5) is a linear combination of the vectors v = (17, 4) and w = (4, 9) given by:
u = 0.75 * v + 1.5 * w = (18.75, 16.5).
Начало формы
Linear Combination
u = s1 * v + s2 * w
v1: 17; v2: 4; w1:4; w2: 9; s1: 0.75; s2: 1.5
Конец формы
Vectors { v 1, v 2,.... v k} in Rn are linearly dependent if and only if one vector can be expressed as a linear combination of the remaining vectors.
The vector v depends linearly on the vector w if v = s * w for some scalar s.
The Span of a Set of Vectors.
Let V be a set of vectors { v 1, v 2,.... v k}. The span of the set of vectors in V, span(V), is the set of all linear combinations of the vectors in V:
span(V) = s1 * v 1 + s2 * v 2 +.... + sk * v k, for all possible sets of scalars {s1, s2,.... sk}.
Basis Vectors.
Unit Coordinate Vectors.
Write e i as the vector in Rn whose components are 0's except for the ith component which is a 1.
The vectors { e 1, e 2,.... e n}, called the unit coordinate vectors, are orthonormal since the vectors satisfy e i * e i = 1, and e i * e j = 0 if i and j are different.
For example, in R3:
e 1 = (1, 0, 0),
e 2 = (0, 1, 0),
e 3 = (0, 0, 1).
The vectors { e 1, e 2,.... e n} in Rn are said to form a basis for Rn, since any vector v = (v1, v2,....., vn) in Rn can be expressed as a linear combination of the { e 1, e 2,.... e n} vectors:
v = v1 * e 1 + v2 * e 2 +.... + vn * e n,
ie the sum of the product of each component of v with the corresponding basis vector.
The linearly independent set of vectors { e 1, e 2,.... e n} is said to span the n dimensional space Rn.
Other Basis Systems.
Theorem: Any basis of Rn consists of exactly n linearly independent vectors in Rn.
Theorem: Any n linearly independent vectors in Rn are a basis for Rn.
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