Linear combinations of vectors

We have already seen these as expressions of the form . In general, a linear combination of vectors is a sum of scalar multiples of some vectors.

We now consider what kind of collection of vectors can be obtained by making all possible linear combinations of given vectors. For instance, for our as chosen above, we can ask what vectors can be written as and this leads to equations . These can be obviously solved for all with the answers . Thus, every vector (of the form ) is a linear combination of .

In general, it can be shown by geometry or algebra that for any two choices of vectors , the same conclusion holds, provided that the chosen vectors don't lie along a common line ! Such vectors are said to be independent and they are also said to span the plane .

Indeed, for two vectors , we can check using Cramer's rule that the vectors span the plane iff the determinant is nonzero, and that in this case, every vector in the plane can be uniquely written as a combination of . Explicitly, the formula simply is .

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If the two vectors did lie on a common line, then would just be a scalar mulitple of and then clearly all linear combinations would end up being just a scalar multiple of . So, they all lie along a single line and cannot cover all the plane!

We can next investigate how certain special linear combinations can given special sets of vectors. For instance, it is easy to show that if is the tip of a vector and is the tip of a vector , then all points along the line segment have position vectors given by the formula where lies in the interval . It is instructive to see the resulting points for special values of like etc.

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> display([DV([0,0],[3,1],``,blue),
PT([3,1],` u`,20),
DV([0,0],[1,4],``,blue),
PT([1,4],` v`,20),
DV([0,0],[5/3,3],``,yellow),PT([5/3,3],` u/3+2v/3`,20),
DV([0,0],[2,5/2],` `,red),PT([2,5/2], ` (u+v)/2`,20),
DL([3,1],[1,4])]);

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