Linear Algebra

38 Questions | Total Attempts: 121

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Linear Algebra

Questions and Answers
  • 1. 
    The dimension of the column space of A is the number of columns in RREF(A) that contain leading ones.
    • A. 

      True

    • B. 

      False

  • 2. 
    The dimension of the column space of A and the null space of A add up to the number of rows in A.
    • A. 

      True

    • B. 

      False

  • 3. 
    If a set of p vectors spans a p-dimensional subspace H of Rn, then these vectors forma basis for H.
    • A. 

      True

    • B. 

      False

  • 4. 
    The dimension of the null space of A is the number of variables (xi) in the equation Ax=0.
    • A. 

      True

    • B. 

      False

  • 5. 
    The dimension of the column space of A is the rank of A.
    • A. 

      True

    • B. 

      False

  • 6. 
    If H is a p-dimensional subspace of Rn, then a linearly independent set of p vectors in H must form a basis for H.
    • A. 

      True

    • B. 

      False

  • 7. 
    A null space is a vector space.
    • A. 

      True

    • B. 

      False

  • 8. 
    The column space of an matrix is in Rm.
    • A. 

      True

    • B. 

      False

  • 9. 
    If u and v are linearly independent, and if w is in Span{u,v} then {u,v,w} is linearly dependent.
    • A. 

      True

    • B. 

      False

  • 10. 
    If three vectors in R3 lie in the same plane in R3, then they are linearly dependent.
    • A. 

      True

    • B. 

      False

  • 11. 
    If a set of vectors from Rn contains fewer than n vectors, then the set is linearly independent.
    • A. 

      True

    • B. 

      False

  • 12. 
    If v1 and v2 are vectors in R4 and v2 is not a scalar multiple of v1 then {v1,v2} is linearly independent.
    • A. 

      True

    • B. 

      False

  • 13. 
    If v1, v2, v3 are in R3 and v3 is not a linear combination of v1, v2, then {v1,v2,v3} is linearly independent.
    • A. 

      True

    • B. 

      False

  • 14. 
    If {v1,..., v4} is a linearly independent set of vectors in R4 then {v1,v2,v3} is also linearly independent.
    • A. 

      True

    • B. 

      False

  • 15. 
    For a nonzero scalar , the vector is times a s long as and has the same direction as
    • A. 

      True

    • B. 

      False

  • 16. 
    The vector (10, 30, -13, 14, -7, 27) can be written as a linear combination of the vectors (1,2,-3,4,-1,2), (1,-2,1,-1,2,1), (0, 2, -1, 2, -1, -1), (1,0,3,-4,1,2), and (1, -2, 1, -1, 2, -3).
    • A. 

      True

    • B. 

      False

  • 17. 
    You can show that a subset of Rn is a subspace of Rn by giving specific numeric examples for which the vectors sum to a vector in the space and for which the negative of the vector is in the space.
    • A. 

      True

    • B. 

      False

  • 18. 
    You can show that a subset of Rn is not a subspace of Rn by giving specific numeric examples for which the vectors do not sum to a vector in the space or for which the negative of the vector is not in the space.
    • A. 

      True

    • B. 

      False

  • 19. 
    If W is a subspace of R2 then W must contain the vector (0,0).
    • A. 

      True

    • B. 

      False

  • 20. 
    If W and U are subspaces of R2 then the union (collection of all vectors in either W or U or both) is also a subspace of R2.
    • A. 

      True

    • B. 

      False

  • 21. 
    Subspace? {(u1,u2) in R2 such that u1u2=0}
    • A. 

      True

    • B. 

      False

  • 22. 
    Subspace? {(u1,u2) in R2 such that u12+u22 is less than or equal to 1}
    • A. 

      True

    • B. 

      False

  • 23. 
    Subspace? {(2s-2, 2s+4t, -t) in R3 such that s and t are real numbers}
    • A. 

      True

    • B. 

      False

  • 24. 
    Subspace? {(2s, 2s+4t, -t) in R3 such that s and t are real numbers}
    • A. 

      True

    • B. 

      False

  • 25. 
    Subspace? {(u1, u2, u3) in R3 such that u1 is equal to u2}
    • A. 

      True

    • B. 

      False