Which ones of the following linear transformations are NOT invertible? (Choose one or more)

Question

BlackTom AI · Accepted Answer

A linear map on R^2 is invertible exactly when it is bijective, equivalently when its 2×2 matrix has nonzero determinant. Geometrically this means the unit square must map to a parallelogram (a 2D region) of nonzero area; any collapse to a line or a point has area zero and is not invertible.

1) Relevant knowledge points
- A linear transformation A: R^2 → R^2 maps vertices (0,0),(1,0),(0,1),(1,1) to 0, v1, v2, v1+v2 where v1 = A(1,0) and v2 = A(0,1).
- A is invertible ⇔ {v1, v2} are linearly independent ⇔ det([v1 v2]) ≠ 0.
- If the image is a point or a 1-dimensional line, det = 0 and A is not invertible.

2) Step-by-step reasoning for each option
a) Image is the single point (0,0). That means A(x) = 0 for all x (the zero map). The matrix is the zero matrix, determinant 0. So A is not invertible.

b) Claimed image vertices: (0,0), (2,0), (1,1), (1,−1). Find v1 and v2 so these match {0, v1, v2, v1+v2}. Let v1 = (1,1), v2 = (1,−1). Then v1+v2 = (2,0). Form matrix with columns v1,v2:
[ [1, 1],
  [1, −1] ].
Compute determinant: 1·(−1) − 1·1 = −1 − 1 = −2 ≠ 0. So v1, v2 are independent and A is invertible. Thus (b) is invertible.

c) Vertices (0,0),(2,0),(0,5),(2,5) correspond to v1=(2,0), v2=(0,5). Matrix:
[ [2, 0],
  [0, 5] ].
Determinant = 2·5 − 0·0 = 10 ≠ 0. So A is invertible.

d) Image is a line segment from (0,0) to (1,3). That means the image lies on the 1-dimensional subspace spanned by (1,3). The two column vectors v1,v2 are collinear (dependent), so det = 0. A collapses area to a line and is not invertible.

3) Calculations shown above:
- b determinant: 1*(−1) − 1*1 = −2 (nonzero).
- c determinant: 2*5 − 0*0 = 10 (nonzero).
- a and d correspond to determinant 0 (zero map or rank 1).

4) Why the correct answers are correct
Options a and d reduce the 2D domain to lower-dimensional images (point or line), so the linear map cannot be bijective and hence is not invertible.

5) Why the other options are incorrect
Option b is incorrect to mark as non-invertible because its implied column vectors are linearly independent (det = −2). Option c is invertible because columns are independent (det = 10).

6) Conclusion
The non-invertible transformations are a and d, which matches the provided answer.

Which ones of the following linear transformations are NOT invertible? (Choose one or more)多项选择题

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