H_
Linear Independence

In each of the problems with square check boxes, you need to check all the correct answers in order for the answer to be marked correct.

SKIP_ p1 QM_[0;;B;C;;E;;] Which of the following sets of vectors are linearly independent?
  1. {(0, 0, 0)}
  2. {(1, 2, 3)}
  3. {(1, 2, 3), (2, 2, 2)}
  4. {(1, 2, 3), (2, 2, 2), (3, 2, 1)}
  5. {(1, 2, 3), (0, 2, 3), (0, 0, 3)}
  6. {(2, -3, 5), (-7, 1, -2), (-7, -18, 29)}
AB_[A;B;C;D;E;F;none of these] SKIP_ p2 QM_[0;A;B;;;E;F;] Which of the following sets of functions in the vector space of polynomials in one variable with real coefficients are linearly independent?
  1. {3x - 2}
  2. images/ma322c2b1.png
  3. images/ma322c2b2.png
  4. images/ma322c2b3.png
  5. {4-13x, 13-4x}
  6. images/ma322c2b4.png
AB_[A;B;C;D;E;F;none of these] SKIP_ p3 QM_[0;A;B;;D;;F;G;] Which of the following sets of functions are linearly independent? (In each case, do you know what the vector space is?)
  1. images/ma322c2b5.png
  2. {tan(x), cot(x)}
  3. images/ma322c2b6.png
  4. images/ma322c2b7.png
  5. images/ma322c2b8.png
  6. {1/x, 1/(x+1)}
  7. {x, |x|}
AB_[A;B;C;D;E;F;G;none of these] SKIP_ p4 QM_[0;;;;D;;;G;] Which of the following are true?
  1. Any set consisting of a single vector is linearly independent.
  2. Any finite set of vectors is linearly independent.
  3. Any non-empty vector subspace is linearly independent.
  4. If S is a set of linearly independent vectors and images/ma322c2b9.png is a vector not in the span of S, then images/ma322c2b10.png is linearly independent.
  5. If S is a set of linearly independent vectors and images/ma322c2b11.png is a vector in the span of S, then images/ma322c2b12.png is never linearly independent.
  6. Two vectors in a vector space are linearly independent if one is not a multiple of the other.
  7. Two vectors in a vector space are linearly independent if neither is a multiple of the other.
AB_[A;B;C;D;E;F;G;none of these] Make sure you can either prove the assertion of give a counterexample. SKIP_ p5 QM_[0;true] Let S be a set of non-zero vectors. If any two vectors in S are perpendicular, then S is linearly independent. AS_[true;false] Make sure you can either prove the assertion of give a counterexample. SKIP_ p6 QM_[0;false] A set of linearly independent vectors can contain a subset which is not linearly independent. AS_[true;false] Make sure you can either prove the assertion or give a counter-example. SKIP_ p7 QM_[0;false] The union of two sets of linearly independent vectors is never linearly independent. AS_[true;false] Make sure you can either prove the assertion or give a counter-example. SKIP_ p8 QM_[0;false] The unit circle defined by images/ma322c2b13.png is a linearly independent set in images/ma322c2b14.png . AS_[true;false] Make sure you can either prove the assertion or give a counter-example. SKIP_ p9 QM_[0;true] If p(x), q(x), and r(x) are three polynomials of degree at most 3 such that {p(x), q(x), r(x)} is linearly independent, then images/ma322c2b15.png is also linearly independent regardless of the choice of real numbers a, b, and c. AS_[true;false] Make sure you can either prove the assertion or give a counter-example. SKIP_ p10 QM_[0;false] A function and its derivative always form a linearly independent set of functions. AS_[true;false] Make sure you can either prove the assertion or give a counter-example. SKIP_