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Cogito

Linear Algebra · Chapter 7 · Lesson 1

Practice Worksheet

Eigenvectors and Eigenvalues

The directions a transformation refuses to turn.

12 problems · about 24 minutes · N-VM.C.11

NameDate

Figure — use these to answer the problems

The dashed green lines are the eigen-directions. Change the entries and watch which directions the transformation still refuses to turn.
ijv
  • i-hat lands on(3, 1)
  • j-hat lands on(1, 3)
  • Determinant8
  • v lands on(4, 4)
  • Eigenvalues4 and 2
a3
b1
c1
d3
v, first entry1
v, second entry1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Av = ⟨9, 12⟩ where v = ⟨3, 4⟩. What is the eigenvalue?

    Answer

  2. What is special about an eigenvector?

    • The transformation leaves it on its own line, only stretching it.
    • The transformation leaves it completely unchanged.

  3. Av = ⟨6, 8⟩ where v = ⟨3, 4⟩. What is the eigenvalue?

    Answer

  4. Av = ⟨−4, −2⟩ where v = ⟨2, 1⟩. What is the eigenvalue?

    Answer

  5. A reflection. What is the eigenvalue for a vector perpendicular to the mirror line?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A scaling by 7 in every direction. What is its eigenvalue?

    Answer

  2. An eigenvalue of 0. Is the matrix singular? 1 yes, 0 no.

    Answer

  3. A 90 degree rotation. How many real eigenvectors does it have?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Match each transformation to an eigenvalue it has.

    Draw a line from each item on the left to its match on the right.

    • The identity
    • A scaling by 3
    • A reflection, perpendicular to the mirror
    • 1
    • 3
    • −1

  2. The identity matrix. What is its eigenvalue?

    Answer

  3. The Stretch

    Av = ⟨10, 15⟩ where v = ⟨2, 3⟩. What is the eigenvalue?

    Answer

  4. The Reflection

    A reflection across a line. What is the eigenvalue for a vector lying along that mirror line?

    Answer