Comment. Linear Transformations quiz
Comment. This quiz is designed to test your knowledge of linear transformations and related concepts
Comment. such as rank, nullity, invertibility, null space, range, etc.
Shuffle Questions.
Shuffle Answers.
Question. Let T: R^4 -> R^4 be the transformation T(x_1,x_2,x_3,x_4) = (0,x_1,x_2,x_3). The null space (or kernel) N(T) of T consists of all vectors of the form
Answer. (0, x_1, x_2, x_3), where x_1, x_2, x_3 are real numbers
Answer. (x_1, 0, 0, 0), where x_1 is a real number
Answer. (1, 0, 0, 0)
Answer. (x_1, x_2, x_3, 0), where x_1, x_2, x_3 are real numbers
Answer. (0, 0, 0, 1)
Correct Answer. (0, 0, 0, x_4), where x_4 is a real number
Answer. (x_4, 0, 0, 0), where x_4 is a real number
Question. Let T: R^2 -> R^2 be the transformation T(x_1,x_2) = (x_1,0). The null space (or kernel) N(T) of T is
Answer. 1
Answer. (x_1, 0)
Answer. {(x_1, 0) : x_1 is real}
Answer. (0, x_2)
Correct Answer. {(0, x_2) : x_2 is real}
Answer. (0, 1)
Answer. (1, 0)
Question. Let T: R^4 -> R^4 be the transformation T(x_1,x_2,x_3,x_4) = (x_2,x_3, 0, 0). The null space (or kernel) N(T) of T consists of all vectors of the form
Answer. (0, 0, x_3, x_4), where x_3 and x_4 are real numbers
Correct Answer. (x_1, 0, 0, x_4), where x_1 and x_4 are real numbers
Answer. (x_2, x_3, 0, 0), where x_2 and x_3 are real numbers
Answer. (0, x_2, x_3, 0), where x_2 and x_3 are real numbers
Answer. (1, 0, 0, 0) and (0, 1, 0, 0)
Answer. (0, 0, x_1, x_4), where x_1 and x_4 are real numbers
Answer. (1, 0, 0, 0) and (0, 0, 0, 1)
Question. Let T: R^3 -> R^4 be the transformation T(x_1, x_2, x_3) = (x_1, 0, x_2, x_2). The range R(T) of T has many bases; one of them is the set of vectors
Correct Answer. (1, 0, 0, 0) and (0, 0, 1, 1)
Answer. (x_1, 0, 0, 0), (0, 0, x_2, 0), and (0, 0, 0, x_2)
Answer. (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1)
Answer. (x_1, 0, 0), (0, x_2, 0), and (0, 0, x_3)
Answer. (1, 0, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1)
Answer. (x_1, 0, 0) and (0, x_2, 0)
Answer. (1, 0, 0) and (0, 1, 0)
Question. Let T: R^3 -> R^4 be the transformation T(x_1, x_2, x_3) = (x_1, 0, x_2, x_2). The null space (or kernel) N(T) of T has many bases; one of them is the set of vectors
Answer. (1, 0, 0, 0) and (0, 0, 1, 1)
Correct Answer. (0, 0, 1)
Partially Correct Answer. (0, 0, x_3)
Answer. (0, 1, 0, 0)
Answer. (x_1, x_2, 0)
Answer. (1, 0, 0) and (0, 1, 0)
Question. Let T: R^4 -> R^4 be the transformation T(x_1,x_2,x_3,x_4) = (0,x_1,x_2,x_3). The image Im(T) of T consists of all vectors of the form
Correct Answer. (0, x_1, x_2, x_3), where x_1, x_2, x_3 are real numbers
Answer. (x_1, x_2, x_3, x_4), where x_1, x_2, x_3, x_4 are real numbers
Answer. (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1)
Answer. (x_1, x_2, x_3, 0), where x_1, x_2, x_3 are real numbers
Answer. (0, 0, 0, x_4), where x_4 is a real number
Answer. (x_4, 0, 0, 0), where x_4 is a real number
Question. A transformation T: R^n -> R^m is linear if and only if
Answer. T is one-to-one and onto.
Comment. This is what it means for T to be invertible.
Correct Answer. There exists a matrix A such that Tx = Ax for all x in R^n.
Answer. The graph of T takes the form y = mx+c.
Comment. This is what it means for T to be affine-linear, not linear. Also, this definition only works in one dimension (unless m is allowed to be a matrix and c is allowed to be a vector).
Partially Correct Answer. One has T(x+y) = Tx + Ty for all vectors x,y.
Correct Answer. One has T(x+y) = Tx+Ty and T(cx)=c T(x) for all vectors x,y and scalars 0.
Answer. No condition required (all transformations are linear).
Answer. The image of T is a line.
Question. If a linear transformation T: R^3 -> R^5 is one-to-one, then
Answer. The rank is three and the nullity is two.
Answer. The situation is impossible.
Answer. The rank is five and the nullity is two.
Answer. The rank is two and the nullity is three.
Correct Answer. The rank is three and the nullity is zero.
Comment. Thanks to blueman for correcting this answer.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to three.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to five.
Question. If a linear transformation T: R^3 -> R^5 is onto, then
Answer. The rank is three and the nullity is two.
Correct Answer. The situation is impossible.
Answer. The rank is five and the nullity is two.
Answer. The rank is two and the nullity is three.
Answer. The rank is three and the nullity is zero.
Comment. Thanks to blueman for correcting this answer.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to three.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to five.
Question. If a linear transformation T: R^5 -> R^3 is onto, then
Answer. The rank is three and the nullity is zero.
Comment. Thanks to blueman for correcting this answer.
Answer. The situation is impossible.
Answer. The rank is five and the nullity is two.
Answer. The rank is two and the nullity is three.
Correct Answer. The rank is three and the nullity is two.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to three.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to five.
Question. If a linear transformation T: R^3 -> R^5 is onto, then
Answer. The rank is three and the nullity is two.
Correct Answer. The situation is impossible.
Answer. The rank is five and the nullity is two.
Answer. The rank is two and the nullity is three.
Answer. The rank is three and the nullity is zero.
Comment. Thanks to blueman for correcting this answer.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to three.
Answer. The rank and nullity can be any pair of non-negative numbers that add up to five.
Question. Let T: R^3 -> R^5 be a linear transformation. Then
Correct Answer. T is one-to-one if and only if the rank is three; T is never onto.
Answer. T is onto if and only if the rank is three; T is never one-to-one.
Answer. T is one-to-one if and only if the rank is two; T is never onto.
Answer. T is onto if and only if the rank is two; T is never one-to-one.
Answer. T is one-to-one if and only if the rank is five; T is never onto.
Answer. T is onto if and only if the rank is five; T is never one-to-one.
Answer. T is invertible if and only if the rank is five.
Question. Let T: R^5 -> R^3 be a linear transformation. Then
Answer. T is one-to-one if and only if the rank is three; T is never onto.
Correct Answer. T is onto if and only if the rank is three; T is never one-to-one.
Answer. T is one-to-one if and only if the rank is two; T is never onto.
Answer. T is onto if and only if the rank is two; T is never one-to-one.
Answer. T is one-to-one if and only if the rank is five; T is never onto.
Answer. T is onto if and only if the rank is five; T is never one-to-one.
Answer. T is invertible if and only if the rank is five.
Question. Let T: R^5 -> R^3 be a linear transformation. Then
Answer. T is one-to-one if and only if the nullity is two; T is never onto.
Correct Answer. T is onto if and only if the nullity is two; T is never one-to-one.
Answer. T is one-to-one if and only if the nullity is zero; T is never onto.
Answer. T is onto if and only if the nullity is zero; T is never one-to-one.
Answer. T is one-to-one if and only if the nullity is three; T is never onto.
Answer. T is onto if and only if the nullity is three; T is never one-to-one.
Answer. T is invertible if and only if the nullity is zero.
Question. Let T: R^3 -> R^5 be a linear transformation. Then
Answer. T is one-to-one if and only if the nullity is two; T is never onto.
Answer. T is onto if and only if the nullity is two; T is never one-to-one.
Correct Answer. T is one-to-one if and only if the nullity is zero; T is never onto.
Answer. T is onto if and only if the nullity is zero; T is never one-to-one.
Answer. T is one-to-one if and only if the nullity is three; T is never onto.
Answer. T is onto if and only if the nullity is three; T is never one-to-one.
Answer. T is invertible if and only if the nullity is zero.