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Quiz: Linear transformations

Page history last edited by Vincent 1 year, 8 months ago

TThis quiz is designed to test your knowledge of linear transformations and related concepts such as rank, nullity, invertibility, null space, range, etc.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula be the transformation Formula. The null space (or kernel) Formula of Formula consists of all vectors of the form
    1. Formula, where Formula are real numbers
    2. Formula, where Formula is a real number
    3. Formula
    4. Formula, where Formula are real numbers
    5. Formula
    6. Formula, where Formula is a real number
      • CORRECT.
    7. Formula, where Formula is a real number
  2. Let Formula be the transformation Formula. The null space (or kernel) Formula of Formula is
    1. Formula
    2. Formula
    3. Formula
    4. Formula
    5. Formula
      • CORRECT.
    6. Formula
    7. Formula
  3. Let Formula be the transformation Formula. The null space (or kernel) Formula of Formula consists of all vectors of the form
    1. Formula, where Formula and Formula are real numbers
    2. Formula, where Formula and Formula are real numbers
      • CORRECT.
    3. Formula, where Formula and Formula are real numbers
    4. Formula, where Formula and Formula are real numbers
    5. Formula and Formula
    6. Formula, where Formula and Formula are real numbers
    7. Formula and Formula
  4. Let Formula be the transformation Formula. The range Formula of Formula has many bases; one of them is the set of vectors
    1. Formula and Formula
      • CORRECT.
    2. Formula, Formula, and Formula
    3. Formula, Formula, Formula, and Formula
    4. Formula, Formula, and Formula
    5. Formula, Formula, and Formula
    6. Formula and Formula
    7. Formula and Formula
  5. Let Formula be the transformation Formula. The null space (or kernel) Formula of Formula has many bases; one of them is the set of vectors
    1. Formula and Formula
    2. Formula
      • CORRECT.
    3. Formula
      • PARTIALLY.
    4. Formula
    5. Formula
    6. Formula and Formula
  6. Let Formula be the transformation Formula. The image Formula of Formula consists of all vectors of the form
    1. Formula, where Formula are real numbers
      • CORRECT.
    2. Formula, where Formula are real numbers
    3. Formula, Formula and Formula
    4. Formula, where Formula are real numbers
    5. Formula, where Formula is a real number
    6. Formula, where Formula is a real number
  7. A transformation Formula is linear if and only if
    1. Formula is one-to-one and onto.
      • INCORRECT. This is what it means for Formula to be invertible.
    2. There exists a matrix Formula such that Formula for all Formula.
      • CORRECT.
    3. The graph of Formula takes the form Formula.
      • INCORRECT. This is what it means for Formula to be affine-linear, not linear. Also, this definition only works in one dimension (unless Formula is allowed to be a matrix and Formula is allowed to be a vector).
    4. One has Formula for all vectors Formula.
      • PARTIALLY.
    5. One has Formula and Formula for all vectors Formula and scalars Formula.
      • CORRECT.
    6. No condition required (all transformations are linear).
    7. The image of Formula is a line.
  8. If a linear transformation Formula is one-to-one, then
    1. The rank is three and the nullity is two.
    2. The situation is impossible.
    3. The rank is five and the nullity is two.
    4. The rank is two and the nullity is three.
    5. The rank is three and the nullity is zero.
      • CORRECT. Thanks to blueman for correcting this answer.
    6. The rank and nullity can be any pair of non-negative numbers that add up to three.
    7. The rank and nullity can be any pair of non-negative numbers that add up to five.
  9. If a linear transformation Formula is onto, then
    1. The rank is three and the nullity is two.
    2. The situation is impossible.
      • CORRECT.
    3. The rank is five and the nullity is two.
    4. The rank is two and the nullity is three.
    5. The rank is three and the nullity is zero.
      • INCORRECT. Thanks to blueman for correcting this answer.
    6. The rank and nullity can be any pair of non-negative numbers that add up to three.
    7. The rank and nullity can be any pair of non-negative numbers that add up to five.
  10. If a linear transformation Formula is onto, then
    1. The rank is three and the nullity is zero.
      • INCORRECT. Thanks to blueman for correcting this answer.
    2. The situation is impossible.
    3. The rank is five and the nullity is two.
    4. The rank is two and the nullity is three.
    5. The rank is three and the nullity is two.
      • CORRECT.
    6. The rank and nullity can be any pair of non-negative numbers that add up to three.
    7. The rank and nullity can be any pair of non-negative numbers that add up to five.
  11. If a linear transformation Formula is one-to-one, then
    1. The rank is three and the nullity is two.
    2. The situation is impossible.
      • CORRECT.
    3. The rank is five and the nullity is two.
    4. The rank is two and the nullity is three.
    5. The rank is three and the nullity is zero.
      • INCORRECT. Thanks to blueman for correcting this answer.
    6. The rank and nullity can be any pair of non-negative numbers that add up to three.
    7. The rank and nullity can be any pair of non-negative numbers that add up to five.
  12. Let Formula be a linear transformation. Then
    1. Formula is one-to-one if and only if the rank is three; Formula is never onto.
      • CORRECT.
    2. Formula is onto if and only if the rank is three; Formula is never one-to-one.
    3. Formula is one-to-one if and only if the rank is two; Formula is never onto.
    4. Formula is onto if and only if the rank is two; Formula is never one-to-one.
    5. Formula is one-to-one if and only if the rank is five; Formula is never onto.
    6. Formula is onto if and only if the rank is five; Formula is never one-to-one.
    7. Formula is invertible if and only if the rank is five.
  13. Let Formula be a linear transformation. Then
    1. Formula is one-to-one if and only if the rank is three; Formula is never onto.
    2. Formula is onto if and only if the rank is three; Formula is never one-to-one.
      • CORRECT.
    3. Formula is one-to-one if and only if the rank is two; Formula is never onto.
    4. Formula is onto if and only if the rank is two; Formula is never one-to-one.
    5. Formula is one-to-one if and only if the rank is five; Formula is never onto.
    6. Formula is onto if and only if the rank is five; Formula is never one-to-one.
    7. Formula is invertible if and only if the rank is five.
  14. Let Formula be a linear transformation. Then
    1. Formula is one-to-one if and only if the nullity is two; Formula is never onto.
    2. Formula is onto if and only if the nullity is two; Formula is never one-to-one.
      • CORRECT.
    3. Formula is one-to-one if and only if the nullity is zero; Formula is never onto.
    4. Formula is onto if and only if the nullity is zero; Formula is never one-to-one.
    5. Formula is one-to-one if and only if the nullity is three; Formula is never onto.
    6. Formula is onto if and only if the nullity is three; Formula is never one-to-one.
    7. Formula is invertible if and only if the nullity is zero.
  15. Let Formula be a linear transformation. Then
    1. Formula is one-to-one if and only if the nullity is two; Formula is never onto.
    2. Formula is onto if and only if the nullity is two; Formula is never one-to-one.
    3. Formula is one-to-one if and only if the nullity is zero; Formula is never onto.
      • CORRECT.
    4. Formula is onto if and only if the nullity is zero; Formula is never one-to-one.
    5. Formula is one-to-one if and only if the nullity is three; Formula is never onto.
    6. Formula is onto if and only if the nullity is three; Formula is never one-to-one.
    7. Formula is invertible if and only if the nullity is zero.

 

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Comments (1)

Areej said

at 6:42 pm on Jul 22, 2020

How to see all 200 question

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