The span of a set of vectors is the collection of all vectors which can be represented by some linear combination of the set. That sounds confusing, but let’s think back to the basis vectors i=(1,0) and j=(0,1) in R^2. If you choose absolutely any vector, anywhere in R^2, you can get to that vector using a linear combination of i and j. If I choose (13,2), I can get to it with the linear combination a=13i+2j, or if I choose (-1,-7), I can get to it with the linear combination a=-i-7j. There’s no vector you can find in R^2 that you can’t reach with a linear combination of i and j.

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