Vector Spaces: The Span Of A Set Of Vectors.

Span{v1 … vp} for suitable vectors

the solution set of a homogeneous equation Ax = 0 can always be expressed explicitly as …

If v1, v2, …, vp are a set of vectors, the span of these vectors, denoted as Span{v1, v2, …, vp}, is the set of all linear combinations of these vectors. In other words, for any scalars c1, c2, …, cp, the vector c1*v1 + c2*v2 + … + cp*vp is in the span of these vectors.

For example, suppose we have two vectors v1 = (1, 2) and v2 = (3, 1). Then, Span{v1, v2} consists of all possible linear combinations of v1 and v2:

– The vector (0, 0) is in Span{v1, v2}, since 0*v1 + 0*v2 = (0, 0).
– The vector (4, 5) is in Span{v1, v2}, since 2*v1 + 1*v2 = (4, 5).
– The vector (-1, 1) is not in Span{v1, v2}, since there are no scalars c1 and c2 such that c1*v1 + c2*v2 = (-1, 1).

In general, the span of p vectors in n-dimensional space is a subspace of dimension at most p. If some of these p vectors are linearly dependent, i.e., one of them can be expressed as a linear combination of the others, then the dimension of their span is less than p. On the other hand, if the p vectors are linearly independent, then the dimension of their span is exactly p.

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