Which ones of the following linear transformations are NOT invertible? (Choose one or more)Question 12Select one or more:a.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into a point (0,0)(0,0).b.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into a square with vertices (0,0)(0,0), (2,0)(2,0), (1,1)(1,1), and (1,−1)(1,−1).c.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into a rectangle with vertices (0,0)(0,0), (2,0)(2,0), (0,5)(0,5), and (2,5)(2,5).d.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into the line segment from (0,0)(0,0) to (1,3)(1,3).
Question
Which ones of the following linear transformations are NOT invertible? (Choose one or more)Question 12Select one or more:a.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into a point (0,0)(0,0).b.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into a square with vertices (0,0)(0,0), (2,0)(2,0), (1,1)(1,1), and (1,−1)(1,−1).c.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into a rectangle with vertices (0,0)(0,0), (2,0)(2,0), (0,5)(0,5), and (2,5)(2,5).d.A𝐴 transforms the unit square [0,1]×[0,1][0,1]×[0,1] into the line segment from (0,0)(0,0) to (1,3)(1,3).
Solution
A linear transformation is invertible if and only if it is bijective, meaning it is both injective (one-to-one) and surjective (onto).
a. A transforms the unit square [0,1]×[0,1] into a point (0,0). This transformation is not invertible because it is not injective. Multiple points from the unit square are mapped to the same point (0,0), so the transformation cannot be reversed.
b. A transforms the unit square [0,1]×[0,1] into a square with vertices (0,0), (2,0), (1,1), and (1,−1). This transformation is invertible because it is both injective and surjective. Each point in the unit square is mapped to a unique point in the transformed square, and every point in the transformed square comes from a point in the unit square.
c. A transforms the unit square [0,1]×[0,1] into a rectangle with vertices (0,0), (2,0), (0,5), and (2,5). This transformation is invertible because it is both injective and surjective. Each point in the unit square is mapped to a unique point in the transformed rectangle, and every point in the transformed rectangle comes from a point in the unit square.
d. A transforms the unit square [0,1]×[0,1] into the line segment from (0,0) to (1,3). This transformation is not invertible because it is not injective. Multiple points from the unit square are mapped to the same point on the line segment, so the transformation cannot be reversed.
So, the transformations that are NOT invertible are a and d.
Similar Questions
Which of the following is/are true? (A) Every linear transformation from R 2 to R 2 maps lines onto points or lines (B) Every surjective linear transformation from R 2 to R 2 maps lines onto lines (C) Every bijective linear transformation from R 2 to R 2 maps pairs of parallel lines to pairs of parallel lines (D) Every bijective linear transformation from R 2 to R 2 maps pairs of perpendicular lines to pairs of perpendicular lines
Consider the matrix . Which of the following statements is correct?Question 1Select one:Select one:a.None of the other statements are correct.b.The matrix is invertible and its determinant is not equal to zero.c.The matrix is not invertible and its determinant is not equal to zero.d.The matrix is not invertible and its determinant is equal to zero.e.The matrix is invertible and its determinant is equal to zero.
(c) Determine the concatenated transformation matrix for translation by vector [1 1] followed by rotation of 45 degrees in 2D square matrix of points given [-1,-1] [1,1] [1,-1] [-1,1]
(b) Assume that {v1, v2, · · · , vk} is a basis of Y . Show that the linear transform T : X → Y from (a) is invertible. (Hint: Use (a) to define a suitable linear transform U : Y → X and show that it is the inverse of T .)
Which sequence of transformations produces an image that is not congruent to the original figure?A.A translation of 6 units to the left followed by a reflection across the x-axisB.A reflection across the x-axis followed by a rotation of 180 counterclockwiseC.A translation of 4 units to the left followed by a dilation of a factor of 3D.A rotation of 90 clockwise followed by a translation of 4 units to the left
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.