09-240/Classnotes for Tuesday October 13

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Replacement Theorem


  1. dim(V) = n
    1. If G generates V then . If also then G is a basis.
    2. If L is linearly dependent then . If also then L is a basis. If also then L can be extended to a basis. Proofs a1. If G has a subset which is a basis then that subset has n elements, so .
      a2. Let be a basis of V, then . Now use replacement with G & L = . Hence, .
      a. From a1 and a2, we know . If then G contains a basis . But , so , and hence G is a basis.

      b. Use replacement with G' being some basis of V. |G| = n.

      If then , so , so generates, so L generates, so L is a basis since it is linearly independent.

      We have that L is basis. If the nagain find such that and generates.

      1. generates V.
      2. . So by part a, is a basis.
  2. If V is finite-dimensional (f.d.) and is a subspace of V, then W is also finite, and . If also dim(W) = dim(V) then W = V and if dim(W) < dim(V) then any basis of W can be extended to a basis of V.

    Proof: Assuming W is finite-dimensional, pick a basis of W; is linearly independent in V so by Corollary 3 of part b, . So span() = V = W so V = W. ...

    Assume W is not finite-dimensional. so pick a such that . So is linearly independent in W, and . Pick . So is linearly dependent and . Pick ... continue in this way to get a sequence where n = dim(V) and is linearly independent. There is a contradiction by Corollary 3.b.

The Lagrange Interpolation Formula

[Aside: because ]


Let be distinct points in .

Let be any points in .

Can you find a polynomial such that ? Is it unique?


Can we find a such that


Solution: Let . (Remember capital pi notation.)