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 = {(1, 1), (1, -1)}, the vector (3, 1) has coordinates (a, b). What is a?

    Answer: ______________

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

    Answer: ______________

  3. The columns of the transition matrix taking C-coordinates to B-coordinates are...

    • 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 (2, 1) with B = {(1, 1), (1, minus 1)}. What is its first standard coordinate?

    Answer: ______________

  5. A map has matrix [[2, 0], [0, 5]] in one basis. What is its trace in any other basis?

    Answer: ______________

  6. With A = [[1, 2], [0, 3]] and P = [[1, 1], [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. The columns of the transition matrix taking C-coordinates to B-coordinates are...

    • The C basis vectors written in B-coordinates
    • The B basis vectors written in C-coordinates
    • The standard basis vectors in order
    • A row of zeros followed by ones

  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 = [[1, 2], [0, 3]] and P = [[1, 1], [0, 1]], what is the (1, 2) entry of P^-1 A P?

    Answer: ______________

LightMySky · lightmysky.comW1-mt_HTGuJhX484-s1

Answer key

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

Same map, new coordinates W1-mt_HTGuJhX484-s1

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