Linear Independence, Span and Basis · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Reaching everywhere with nothing spare

Mathematics · Linear Algebra · ages 18-19
Name ______________________   Date ____________

  1. Mia says (1, 1) and (minus 1, minus 1) are independent because neither is the zero vector. Is Mia right?

    Circle one:   True   False

  2. (1, 0, 0) and (0, 1, 0) span a plane. For v = (2, 5, c), which value of c leaves v in that plane, so the three vectors fail to span all of space?

    Answer: ______________

  3. The equation c1*(1, 1) + c2*(2, 2) = (0, 0) holds with c2 = 4. What is c1?

    Answer: ______________

  4. Which of these pairs of vectors is independent in the plane?

    • (1, 2) and (2, 4)
    • (1, 0) and (0, 1)
    • (3, 6) and (1, 2)

  5. Suppose a(1, 1) + b(1, -1) and c(1, 1) + d(1, -1) both equal (4, 2). Subtracting gives (a - c)(1, 1) + (b - d)(1, -1) = (0, 0). What must a - c be?

    • 0, by independence of the basis
    • Any value, since a and c can be chosen freely
    • 1, because (1, 1) is the first basis vector
    • 4, the first entry of (4, 2)

  6. What is the largest possible size of an independent set in the plane?

    • 2
    • 3
    • 4

  7. What is the span of (1, 2) and (3, 6) in the plane?

    • The whole plane
    • A line through the origin
    • Only the two points themselves
    • A line that misses the origin

  8. Why can the vector (6, 2) be written in only one way using (1, 0) and (0, 1)?

    • Because (1, 0) and (0, 1) are independent
    • Because coordinates must be whole numbers
    • Because (6, 2) is longer than each of them

  9. Solve a(1, 2, 0) + b(0, 1, 1) + c(1, 3, 1) = (0, 0, 0) with c = 1. What is a?

    Answer: ______________

  10. Nadia says the set (1, 2), (2, 4), (0, 1) is a basis of the plane because it spans. What is wrong?

    • It does not span the plane
    • (2, 4) is twice (1, 2), so the set is dependent
    • Three vectors can never span the plane
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Reaching everywhere with nothing spare W1-mt_5RGbip9yC9-s1

  1. False · Avoiding the zero vector is not enough: the second is minus 1 times the first.
  2. 0 · The first two vectors span the plane of points with third entry 0. If c = 0, v lies in that plane and depends on them, so the span stays a plane.
  3. -8 · Each component gives c1 + 2c2 = 0, so c1 = -8 when c2 = 4.
  4. (1, 0) and (0, 1) · Only the standard basis pair is independent; each other pair holds a multiple.
  5. 0, by independence of the basis · (1, 1) and (1, -1) are independent, so the only combination giving zero is the trivial one. So a - c = 0, and the coordinates are unique.
  6. 2 · Two independent vectors already span the plane, so a third one always depends on them.
  7. A line through the origin · (3, 6) is 3 times (1, 2), so every combination is a multiple of (1, 2). Those multiples fill the line through the origin in that direction.
  8. Because (1, 0) and (0, 1) are independent · Two different expressions would differ by a combination giving zero, and independence forces that combination to be trivial.
  9. -1 · The entries give a + c = 0, 2a + b + 3c = 0 and b + c = 0. With c = 1, a = -1 and b = -1, and 2(-1) - 1 + 3 = 0 checks. The set is dependent.
  10. (2, 4) is twice (1, 2), so the set is dependent · Two vectors already span the plane, so the third depends on them, and a basis needs independence.
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