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. Rotation by 180 degrees sends (2, 5) to (a, b). What is a?

    Answer: ______________

  2. What is the matrix of reflection in the y-axis?

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

  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, z) = (x + y + z, 0) maps R cubed to R squared. What is the dimension of its kernel?

    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 map is linear?

    • T(x, y) = (x + 2, y)
    • T(x, y) = (x squared, y)
    • T(x, y) = (3x, x + y)
    • T(x, y) = (xy, x)

  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. The matrix of T is [[1, 2], [2, 4]]. Which describes its image?

    • All of R squared
    • The line through the origin and (1, 2)
    • The line through the origin and (2, 1)
    • Only the zero vector
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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. -2 · A half turn sends (1, 0) to (-1, 0) and (0, 1) to (0, -1), so (2, 5) goes to (-2, -5).
  2. [[-1, 0], [0, 1]] · (1, 0) lands on (-1, 0) and (0, 1) stays at (0, 1). Those images are the columns, giving rows (-1, 0) and (0, 1).
  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. 2 · Every output is a multiple of (1, 0), so the image is 1D. The kernel has dimension 3 minus 1, which is 2.
  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(x, y) = (3x, x + y) · (3x, x + y) keeps sums and scalings. Adding 2 moves zero, squaring breaks scaling, and xy breaks scaling.
  9. [[0, 1], [1, 0]] · (1, 0) lands on (0, 1) and (0, 1) lands on (1, 0); stack those as columns.
  10. The line through the origin and (1, 2) · The image is the column space. The columns (1, 2) and (2, 4) both lie on the line through (1, 2).
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