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An infinite set of vectors is linearly independent if every nonempty finite subset is linearly independent. Conversely, an infinite set of vectors is linearly dependent if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.
The elements of a basis are called basis vectors. Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. [1] In other words, a basis is a linearly independent spanning set.
[note 3] If v 1, ..., v k are linearly independent, then the coordinates t 1, ..., t k for a vector in the span are uniquely determined. A basis for a subspace S is a set of linearly independent vectors whose span is S. The number of elements in a basis is always equal to the geometric dimension of the subspace.
In particular, the vectors are linearly independent if and only if the parallelotope has nonzero n-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is nonsingular. When n > m the determinant and volume are zero.
The elements of a subset G of a F-vector space V are said to be linearly independent if no element of G can be written as a linear combination of the other elements of G. Equivalently, they are linearly independent if two linear combinations of elements of G define the same element of V if and only if they have the same coefficients. Also ...
A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i. A set of vectors that spans a vector space is called a spanning set or generating set.
What happens after an executive order is signed? After a president signs an executive order, the White House sends the document to the Office of the Federal Register, the executive branch's ...
By Zorn's lemma, every linearly independent set is contained in a maximal linearly independent set K. This maximality implies that K spans V and is therefore a basis (the maximality implies that every element of V is linearly dependent from the elements of K, and therefore is a linear combination of elements of K).