Web Reference: In this section we discuss subspaces of R n. A subspace turns out to be exactly the same thing as a span, except we don’t have a particular set of spanning vectors in mind. This change in perspective is quite useful, as it is easy to produce subspaces that are not obviously spans. We can consider the xy-plane as the set of all vectors that arise as a linear combination of the two vectors in U. Call this set of all linear combinations the span of U: The span of a set in the smallest subspace containing the set. So, the span of a subspace is the subspace itself.
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