Linear Maps and Their Matrices · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Matrices are maps on basis vectors

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

  1. Projection onto the x-axis sends (3, 5) to (?, 0). What is the missing first entry?

    Answer: ______________

  2. What is the matrix of counterclockwise rotation by 90 degrees?

    • [[0, -1], [1, 0]]
    • [[0, 1], [-1, 0]]
    • [[1, 0], [0, 1]]
    • [[-1, 0], [0, -1]]

  3. What is the matrix of counterclockwise rotation by 90 degrees?

    • [[0, 1], [minus 1, 0]]
    • [[0, minus 1], [1, 0]]
    • [[1, 0], [0, 1]]

  4. A map T sends the zero vector to (1, 0). Can T be linear?

    Circle one:   True   False

  5. T(x, y) = (x + y, 2x + 2y) has a 1D kernel. What is the dimension of its image?

    Answer: ______________

  6. T(x, y, z) = (x + y, z) maps R cubed to R squared and is onto. What is the dimension of its kernel?

    • 0
    • 2
    • 1

  7. Which property must every linear map T satisfy?

    • T(u + v) = T(u) + T(v) for all u, v
    • T(v) is never the zero vector
    • T(c v) = T(c) times T(v) for all scalars c

  8. Which property must every linear map T satisfy?

    • T(u + v) = T(u) + T(v) for all u, v
    • T(u + v) = T(u) * T(v) for all u, v
    • T(v) is never the zero vector
    • T(c v) = T(c) * T(v) for all scalars c

  9. What is the matrix of the reflection in the line y = x?

    • [[0, 1], [1, 0]]
    • [[1, 0], [0, minus 1]]
    • [[0, minus 1], [minus 1, 0]]

  10. T(x, y, z) = (x + y, z) maps R^3 to R^2. What is the dimension of its kernel?

    • 0
    • 1
    • 2
    • 3
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Answer key

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

Matrices are maps on basis vectors W1-mt_pzrfiQFdto-s1

  1. 3 · Projection onto the x-axis keeps x and kills y, so (3, 5) goes to (3, 0).
  2. [[0, -1], [1, 0]] · Rotation sends (1, 0) to (0, 1) and (0, 1) to (-1, 0); these images are the columns.
  3. [[0, minus 1], [1, 0]] · Rotation sends (1, 0) to (0, 1) and (0, 1) to (minus 1, 0); these images are the columns.
  4. False · No: every linear map sends zero to zero, so this T fails the first test.
  5. 1 · Domain is R^2 and kernel takes 1 dimension, so the image has 2 - 1 = 1.
  6. 1 · The map is onto R squared, so rank 2 and kernel dimension = 3 minus 2 = 1.
  7. T(u + v) = T(u) + T(v) for all u, v · Additivity is one of the two defining properties of linearity.
  8. T(u + v) = T(u) + T(v) for all u, v · Additivity is one of the two defining properties of linearity.
  9. [[0, 1], [1, 0]] · (1, 0) lands on (0, 1) and (0, 1) lands on (1, 0); stack those as columns.
  10. 1 · The map is onto R^2, so rank 2 and kernel dimension = 3 - 2 = 1.
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