Answer :
Answer: . Rotation about the y-axis by π
Step-by-step explanation:
The invertible linear transformations from R3 to R3 from the given list are Reflection in the origin, Rotation about the y-axis by
, Dilation by a factor of 4, and the Identity transformation.
A linear transformation from R3 to R3 is considered invertible if it has a two-sided inverse, which means there exists a transformation that can reverse the effect of the original transformation. For a transformation to be linear, it must transform lines into lines and maintain parallelism of lines. Let's examine the transformations listed in the question:
Projection onto the xy-plane (A): This is not invertible because all points with different z-values will be mapped to the same point in the xy-plane, losing the third dimension information.
Reflection in the origin (B): This is invertible, as applying the reflection twice returns each point back to its original position.
Projection onto the z-axis (C): Similar to the projection onto the xy-plane, this is not invertible because information in the xy-plane is lost.
Rotation about the y-axis by π (D): This is an invertible transformation, since rotating back by π will return the points to their original positions.
Dilation by a factor of 4 (E): Dilation is invertible because one can perform a contraction by a factor of 1/4 to undo the dilation.
Identity transformation: T(v)=v for all v (F): This is obviously invertible because it is its own inverse.
Therefore, the invertible linear transformations from the given list are Reflection in the origin (B), Rotation about the y-axis by π (D), Dilation by a factor of 4 (E), and the Identity transformation (F).
