Question

Select all of the linear transformations from ℝ3 to ℝ3 that are invertible. There may be more than one correct answer.

A. Identity transformation (i.e. T(v⃗ )=v⃗ for all v⃗ )
B. Projection onto the xz-plane
C. Reflection in the y-axis
D. Rotation about the x-axis
E. Dilation by a factor of 6
F. Projection onto the z-axis

Answer 1

To determine which linear transformations from ℝ3 to ℝ3 are invertible, we need to check their determinants. The identity transformation, reflection in the y-axis, rotation about the x-axis, and dilation by a factor of 6 are all invertible.

Step-by-step explanation:

A linear transformation from ℝ3 to ℝ3 is invertible if and only if its determinant is non-zero. Let's analyze each option:

A. Identity transformation: The identity transformation is defined as T(v⃗) = v⃗ for all v⃗. Its determinant is 1, so it is invertible.

B. Projection onto the xz-plane: The projection onto the xz-plane has determinant 0, so it is not invertible.

C. Reflection in the y-axis: The reflection in the y-axis has determinant -1, so it is invertible.

D. Rotation about the x-axis: The rotation about the x-axis has determinant 1, so it is invertible.

E. Dilation by a factor of 6: The dilation by a non-zero factor has determinant 6^3 = 216, so it is invertible.

F. Projection onto the z-axis: The projection onto the z-axis has determinant 0, so it is not invertible.