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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.
If that is possible, then v 1,...,v n are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors. If S is linearly independent and the span of S equals V, then S is a basis for V.
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.
A linearly independent set L is a basis if and only if it is maximal, that is, it is not a proper subset of any linearly independent set. If V is a vector space of dimension n, then: A subset of V with n elements is a basis if and only if it is linearly independent. A subset of V with n elements is a basis if and only if it is a spanning set of V.
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 ...
In signal processing, it is common to represent signals as vectors in a Hilbert space. In this interpretation, a vector expressed as a linear combination of the frame vectors is a redundant signal. Representing a signal strictly with a set of linearly independent vectors may not always be the most compact form. [13]
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.
The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space. For example, consider the matrix