In linear algebra, the span of a set of vectors is the set of all linear combinations of those vectors. In other words, it is the set of all possible combinations of the given vectors multiplied by any scalar value.
Understanding the concept of span is crucial in linear algebra as it allows us to determine whether a vector can be expressed as a linear combination of other vectors. It also helps us to understand the relationships between different vectors and their subspaces.
This equation states that the span of the intersection of two sets of vectors is a subset of the intersection of the spans of those two sets. This is important because it shows that the span of the intersection is limited by the span of the individual sets, and it helps us to better understand the relationships between subspaces.
Say we have two sets of vectors, S1 = {(1,2,3), (4,5,6)} and S2 = {(2,4,6), (8,10,12)}. The intersection of these two sets is {(2,4,6)}, and the span of this intersection is a subset of the span of S1 and S2, which is {(2,4,6), (4,8,12), (6,12,18), (8,16,24)}. Therefore, the equation "Span(S1 ∩ S2) ⊆ span(S1) ∩ span(S2)" holds true.
This equation is used in various areas of mathematics and science, such as in solving systems of linear equations and in analyzing data in fields such as statistics and machine learning. It helps to simplify calculations and provides a better understanding of the relationships between different vectors and subspaces.