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

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(Key: correct, incorrect, partially correct.)

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

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