Linalg Midterm Review

• wrong list:
  • def of span (verb)
  • def of a field
  • def of a direct sum
  • sum of subspaces
  • cross product
  • connection between linear independence and systems of equations
  • geometric interpretation of dot product
  • U1 + U2 is a direct sum iff U1 intersect U2 = {0}
  • def vector space
  • elementary matrix
  • solving matrix equations
  • finding inverse
  • find the plane containing
  • prove that a set of vectors is linearly dependent if and only if you can write on of the vectors as a linear combination of the others
  • prove or give a counterexample: v1, v2, v3, v4, basis of V, and U is subspace of V.. del v3 v4 is it a basis?
• solutions
  • def of span (verb)
    • if the span of a list of vectors equals V, then the list of vecs spans V
      • ie. if it contains all the nessasarry info, then it spans.
  • def of a field
    • a set containing at least two distinct elements 0 and 1, along with the operations + and * as defined on the reals/complexs.
      • commutativity
      • associativity
      • additive identity
      • multiplicative identity
      • additive inverse
      • multiplicative inverse
      • distributive property
  • def direct sum
    • sum of subspaces where each element in the resultant subspace can be written uniquely as a sum of the elements in the original subspaces
  • sum of subspaces
    • subspaces containing the set of all possible linear combinations from the union of the original subspaces
  • cross product #TODO!
    • the determinant thingy, or
    • \(|A||B| \sin \theta n\)
      • n is the unit vector orthogonal to vectors A and B
  • connect between linear independence and systems of equations -"if we take the coefficients of a system of equations as vectors, then the vectors are linearly independent if the system has one solution, and linearly dependent if the system has either zero or infinite solutions"
  • geometric interpretation of dot product
    • %%projection of one vector onto the other times the magnitude of the vector %%
    • magnitude of projection of a vector onto another vector times the magnitude of the other vector
      • \(|A||B| \cos \theta\)
  • def vector space
    • set V with addition and scalar multiplication such that there is
      • additive identity
      • additive inverse (no multiplicative inverse!)
      • commutativity
      • assosiativty
      • distibutive property
  • elementary matrix
    • identity matrix with one row operation applied

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