Change of Basis · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Same map, new coordinates

Mathematics · Linear Algebra · ages 19-20
Name ______________________   Date ____________

  1. In basis B = {(2, 0), (0, 3)}, what is the first B-coordinate of (8, 6)?

    Answer: ______________

  2. In basis B = {(1, 1), (1, minus 1)}, the vector (3, 1) has coordinates (a, b). What is a?

    Answer: ______________

  3. What are the columns of the transition matrix that takes C-coordinates to B-coordinates?

    • The B basis vectors written in C-coordinates
    • The C basis vectors written in B-coordinates
    • The standard basis vectors in order

  4. A vector has B-coordinates (4, 1) with B = {(1, 1), (1, minus 1)}. What is its first standard coordinate?

    Answer: ______________

  5. A map has matrix [[1, 4], [2, 3]] in the standard basis. What is its determinant in any other basis?

    Answer: ______________

  6. With A = [[1, 1], [0, 2]] and P = [[1, 2], [0, 1]], what is the (1, 2) entry of P inverse A P?

    Answer: ______________

  7. Changing the basis changes the determinant of the matrix of a fixed map. True or false?

    Circle one:   True   False

  8. B = {(1, 0), (1, 1)}. Which vector has B-coordinates (2, 3)?

    • (2, 3)
    • (3, 5)
    • (5, 3)
    • (5, 5)

  9. A map has eigenvalues 2 and 5 in one basis. What are they in a new basis?

    • still 2 and 5
    • doubled to 4 and 10
    • both become 7

  10. With A = [[2, 0], [0, 1]] and P = [[1, 1], [0, 1]], what is the (1, 2) entry of P inverse A P?

    Answer: ______________

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Answer key

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

Same map, new coordinates W1-mt_HTGuJhX484-s1

  1. 4 · (8, 6) = 4(2, 0) + 2(0, 3), so the B-coordinates are (4, 2).
  2. 2 · a + b = 3 and a minus b = 1 give a = 2, b = 1.
  3. The C basis vectors written in B-coordinates · Each column says where one old basis vector lands in the new coordinate language.
  4. 5 · 4 times (1, 1) plus 1 times (1, minus 1) is (5, 3), so the first standard coordinate is 5.
  5. -5 · The determinant is 1 times 3 minus 4 times 2, which is -5, and a basis change keeps it.
  6. -1 · P inverse is [[1, minus 2], [0, 1]]. P inverse A P works out to [[1, minus 1], [0, 2]], so the (1, 2) entry is minus 1.
  7. False · False: the determinant belongs to the map, so every matrix of the same map shares it.
  8. (5, 3) · 2(1, 0) + 3(1, 1) = (5, 3).
  9. still 2 and 5 · Eigenvalues belong to the map, so a new basis keeps them at 2 and 5.
  10. 1 · P inverse is [[1, -1], [0, 1]]. P inverse A = [[2, -1], [0, 1]], and multiplying by P gives [[2, 1], [0, 1]].
Worksheet · LightMySky