Dimension and the Rank-Nullity Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Rank plus nullity equals columns

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

  1. A 3 by 5 matrix has rank 2. How many free parameters does Ax = 0 have?

    • 2
    • 3
    • 5
    • 0

  2. A 4 by 5 matrix has rank 3. What is its nullity?

    Answer: ______________

  3. A 4 by 5 matrix has rank 3. What is its nullity?

    Answer: ______________

  4. A null space is spanned by (1, 0, -1) and (0, 1, 2). Neither is a multiple of the other. What is the nullity?

    Answer: ______________

  5. Tom says the column space of [[1, 2], [2, 4]] is all of R squared. Is Tom right?

    Circle one:   True   False

  6. Tom says the column space of [[1, 2], [2, 4]] is all of R^2. Is Tom right?

    Circle one:   True   False

  7. A 6 by 6 matrix has rank 6. What does that say about Ax = b?

    • It has exactly one solution for every b
    • It has infinitely many solutions for every b
    • It has no solution for any b
    • It has exactly 6 solutions for every b

  8. A 6 by 6 matrix has rank 6. What does that say about Ax equals b?

    • It has exactly one solution for every b
    • It has infinitely many solutions for every b
    • It has no solution for some b

  9. Can a 4 by 7 matrix have rank 5?

    • Yes, rank 5 is possible here
    • Yes, rank 5 is forced for wide matrices
    • No, the rank cannot pass the 4 rows

  10. A matrix has 5 columns and nullity 3. How many vectors are in a basis for its column space?

    Answer: ______________

LightMySky · lightmysky.comW1-mt_aMG8PcwPAR-s1

Answer key

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

Rank plus nullity equals columns W1-mt_aMG8PcwPAR-s1

  1. 3 · Nullity = 5 - 2 = 3, and each unit of nullity is one free parameter.
  2. 2 · Rank plus nullity equals the 5 columns, so nullity = 5 - 3 = 2.
  3. 2 · Rank plus nullity equals the 5 columns, so nullity is 5 minus 3.
  4. 2 · The two spanning vectors are independent, so they form a basis of size 2.
  5. False · Tom is wrong: the second column is twice the first, so the column space is just a line.
  6. False · Tom is wrong: the second column is twice the first, so the column space is just a line.
  7. It has exactly one solution for every b · Full rank means nullity 0, so solutions are unique, and 6 pivots hit every b.
  8. It has exactly one solution for every b · Full rank means nullity 0, so solutions are unique, and 6 pivots reach every b.
  9. No, the rank cannot pass the 4 rows · Rank is bounded by rows and columns alike, so at most 4 here.
  10. 2 · Rank is 5 minus 3, which is 2, and the rank is the size of a column space basis.
Worksheet · LightMySky