Search results
Results From The WOW.Com Content Network
The third "5 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary to define a specific location on a plane. Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set.
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. Any spanning set for a subspace can be changed into a basis by removing redundant vectors (see § Algorithms below for more). Example
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 ...
Consider the restrictions on x 1, x 2, y 1, y 2 required to make u and v form an orthonormal pair. ... Every orthonormal list of vectors is linearly independent.
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.
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).
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.
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.