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Cogito

Linear Algebra · Chapter 4 · Lesson 2

Practice Worksheet

The Standard Transformations

Rotations, reflections, shears and scalings.

12 problems · about 23 minutes · N-VM.C.11, G-CO.A.2, G-CO.A.5

NameDate

Figure — use these to answer the problems

Set the entries to 1, 0, 0, −1 for a reflection and note the determinant turns negative. Then try 1, 1, 0, 1 for a shear.
ijv
  • i-hat lands on(1, 0)
  • j-hat lands on(0, -1)
  • Determinant-1
  • v lands on(2, -2)
a1
b0
c0
d-1
v, first entry2
v, second entry2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. A scaling with rows [6, 0] and [0, 2]. What is its determinant?

    Answer

  2. What does a negative determinant mean?

    • The transformation has flipped the plane over, reversing orientation.
    • The transformation shrinks everything to nothing.

  3. Rows [2, 0] and [0, 5]. What is the determinant?

    Answer

  4. A reflection. Is its determinant negative? 1 yes, 0 no.

    Answer

  5. A rotation. What is its determinant?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Rows [1, 7] and [0, 1]. What is the determinant?

    Answer

  2. A rotation by 0 degrees. What is the entry in row 1, column 1?

    Answer

  3. A scaling by 4 in both directions. By what factor does area grow?

    Answer

Stretch Yourself

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

4 problems

  1. Match each matrix to the transformation it performs.

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

    • Rows [1, 0] and [0, −1]
    • Rows [1, 3] and [0, 1]
    • Rows [2, 0] and [0, 2]
    • Reflection across the x-axis
    • Shear
    • Uniform scaling

  2. Rows [0, −1] and [1, 0]. What is the determinant?

    Answer

  3. The Double

    A scaling with rows [3, 0] and [0, 3]. What is its determinant?

    Answer

  4. The Shear

    A shear with rows [1, 4] and [0, 1]. What is its determinant?

    Answer