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    Linear Algebra: Spanning sets and basis

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    Could you clarify what constitutes a spanning set and a basis? Also how does one test to see if a set of vectors is a spanning set and if it is a basis?

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    Solution Preview

    Let S be a set of vectors defined over certain field (say, the real numbers).
    Now we can use these vectors to write a new one by linearly adding them together:
    These new vectors form a vector space W
    and the set S is called the "spanning set" of W.

    For example, look at the set in :
    Then all the vectors that can be created from these three vectors are of the form:
    If we look at these vectors as Cartesian coordinates in real space, these vectors all lie in the xy plane.

    Not only that, any vector in the xy plane can be written as a linear combination of these vectors. Let be an arbitrary vector in the xy plane

    So once we choose an arbitrary value for , equation (1.5) tells us what should be the values of the other two coefficients that will "manufacture" the vector w.
    Hence the set is a spanning set of W.
    Obviously, we can have many spanning sets for the same vector space
    We could say that
    Is also a spanning set of W (vectors in the xy plane)

    again we see that we can write any vector in the xy plane as a linear combination of the vectors of equation (1.6) in this spanning set.

    If we want to express the vector in terms of we get:

    Solution Summary

    The expert examines spanning sets and basis for linear algebra.