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Quiz: Linear transformations
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last edited
by Vincent 5 years, 3 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.)
- Let be the transformation . The null space (or kernel) of consists of all vectors of the form
- , where are real numbers
- , where is a real number
- , where are real numbers
- , where is a real number
- , where is a real number
- Let be the transformation . The null space (or kernel) of is
-
- Let be the transformation . The null space (or kernel) of consists of all vectors of the form
- , where and are real numbers
- , where and are real numbers
- , where and are real numbers
- , where and are real numbers
- and
- , where and are real numbers
- and
- Let be the transformation . The range of has many bases; one of them is the set of vectors
- and
- , , and
- , , , and
- , , and
- , , and
- and
- and
- Let be the transformation . The null space (or kernel) of has many bases; one of them is the set of vectors
- and
-
-
- and
- Let be the transformation . The image of consists of all vectors of the form
- , where are real numbers
- , where are real numbers
- , and
- , where are real numbers
- , where is a real number
- , where is a real number
- A transformation is linear if and only if
- is one-to-one and onto.
- INCORRECT. This is what it means for to be invertible.
- There exists a matrix such that for all .
- 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).
- One has for all vectors .
- One has and for all vectors and scalars .
- No condition required (all transformations are linear).
- The image of is a line.
- If a linear transformation is one-to-one, then
- The rank is three and the nullity is two.
- The situation is impossible.
- The rank is five and the nullity is two.
- The rank is two and the nullity is three.
- The rank is three and the nullity is zero.
- CORRECT. Thanks to blueman for correcting this answer.
- The rank and nullity can be any pair of non-negative numbers that add up to three.
- The rank and nullity can be any pair of non-negative numbers that add up to five.
- If a linear transformation is onto, then
- The rank is three and the nullity is two.
- The situation is impossible.
- The rank is five and the nullity is two.
- The rank is two and the nullity is three.
- The rank is three and the nullity is zero.
- INCORRECT. Thanks to blueman for correcting this answer.
- The rank and nullity can be any pair of non-negative numbers that add up to three.
- The rank and nullity can be any pair of non-negative numbers that add up to five.
- If a linear transformation is onto, then
- The rank is three and the nullity is zero.
- INCORRECT. Thanks to blueman for correcting this answer.
- The situation is impossible.
- The rank is five and the nullity is two.
- The rank is two and the nullity is three.
- The rank is three and the nullity is two.
- The rank and nullity can be any pair of non-negative numbers that add up to three.
- The rank and nullity can be any pair of non-negative numbers that add up to five.
- If a linear transformation is one-to-one, then
- The rank is three and the nullity is two.
- The situation is impossible.
- The rank is five and the nullity is two.
- The rank is two and the nullity is three.
- The rank is three and the nullity is zero.
- INCORRECT. Thanks to blueman for correcting this answer.
- The rank and nullity can be any pair of non-negative numbers that add up to three.
- The rank and nullity can be any pair of non-negative numbers that add up to five.
- Let be a linear transformation. Then
- is one-to-one if and only if the rank is three; is never onto.
- is onto if and only if the rank is three; is never one-to-one.
- is one-to-one if and only if the rank is two; is never onto.
- is onto if and only if the rank is two; is never one-to-one.
- is one-to-one if and only if the rank is five; is never onto.
- is onto if and only if the rank is five; is never one-to-one.
- is invertible if and only if the rank is five.
- Let be a linear transformation. Then
- is one-to-one if and only if the rank is three; is never onto.
- is onto if and only if the rank is three; is never one-to-one.
- is one-to-one if and only if the rank is two; is never onto.
- is onto if and only if the rank is two; is never one-to-one.
- is one-to-one if and only if the rank is five; is never onto.
- is onto if and only if the rank is five; is never one-to-one.
- is invertible if and only if the rank is five.
- Let be a linear transformation. Then
- is one-to-one if and only if the nullity is two; is never onto.
- is onto if and only if the nullity is two; is never one-to-one.
- is one-to-one if and only if the nullity is zero; is never onto.
- is onto if and only if the nullity is zero; is never one-to-one.
- is one-to-one if and only if the nullity is three; is never onto.
- is onto if and only if the nullity is three; is never one-to-one.
- is invertible if and only if the nullity is zero.
- Let be a linear transformation. Then
- is one-to-one if and only if the nullity is two; is never onto.
- is onto if and only if the nullity is two; is never one-to-one.
- is one-to-one if and only if the nullity is zero; is never onto.
- is onto if and only if the nullity is zero; is never one-to-one.
- is one-to-one if and only if the nullity is three; is never onto.
- is onto if and only if the nullity is three; is never one-to-one.
- is invertible if and only if the nullity is zero.
Score:
.
Quiz: Linear transformations
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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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