Calculus homework help

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  1. Matrices
    A) Find the LU factorization of the matrix
    B) What is the rank?
    C) Find as basis for the Column Space.
    D) Find a basis for the Row Space.
    E) Find a basis for the Null Space.
    F) What is the Nullity?
  2. Describe the subspaces of R3 geometrically.
  3. Are the set of vectors linearly independent?
  4. Does the following set span ℝ4?
  5. Does the following set span ℝ3?
  6. Does the following set span ℝ3?
  7. Is the vector in Null Space of the matrix ?
  8. Find the orthogonal projection of the vector <5,5,7> on the plane spanned by the vectors {<1,4,8>,<7,1,2>}
  9. Consider the Transformation T(x) = Mx, where M is the matrix .
    Is the linear transformation onto? Is it one-to-one?
  10. T is a linear transformation such that . Find the matrix M that describes T.
  11. A)
    B) Two Pivot columns, so Rank is two!
    C) One possible Basis for the Column Space:
    D) One possible Basis for the Row Space:
    E) One possible Basis for the Null-Space:
    F) Nullity is 2, since the Null-Space is two dimensional.
  12. Trivial Subspace of just the zero vector, lines through the origin, planes through the origin, and the improper subspace of all ℝ­3 itself.
  13. No. They are linearly dependent.
  14. No. You need at least 4 vectors to span ℝ4.
  15. No. Even though there are 3 vectors in the set, there are only 2 independent vectors.
  16. Yes, this set spans ℝ3. However, this set is NOT independent, so it is NOT a basis.
  17. Yes. Multiply to get the zero vector.-3,0,
  18. <-3,0,-3>
  19. Neither. The rank is 1. Since there are 3 columns but the dimension of the column space is 1, it is NOT 1-to-1. Since there are 2 rows, but the dimension of the row space is 1, it is NOT onto.

 
 
 
 
 
 

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