Final answer:
The invertible linear transformations from R³ to R³ are A) Reflection in the origin, C) Identity transformation, E) Rotation about the z-axis, and F) Dilation by a factor of 7.
Step-by-step explanation:
To determine whether the given linear transformations from R³ to R³ are invertible, we need to consider each transformation separately:
- Reflection in the origin: This is an invertible operation because it can be undone by reflecting back through the origin.
- Trivial transformation (T(v)=0 for all v): This is not invertible because there is no unique pre-image for the zero vector (all vectors map to zero).
- Identity transformation (T(v)=v for all v): This is certainly invertible because each vector maps to itself.
- Projection onto the y-axis: This is not invertible because information along the x and z axes is lost during the projection.
- Rotation about the z-axis: This is an invertible operation since rotations can be reversed by rotating in the opposite direction.
- Dilation by a factor of 7: Dilation is invertible as long as the factor isn't zero, which in this case, it's 7, so it's invertible.
In summary, transformations A, C, E, and F are invertible.