# Thread: Linear Transformations and Invertibility

1. ## Linear Transformations and Invertibility

Which of the following linear transformations from R^3 to R^3 are invertible ?

A. Projection onto the z-axis
C. Dilation by a factor of 3
D. Reflection in the origin
E. Trivial transformation (i.e. T(v)=0 for all v)
F. Identity transformation (i.e. T(v)=v for all v)

2. Originally Posted by My Little Pony
Which of the following linear transformations from R^3 to R^3 are invertible ?

A. Projection onto the z-axis
C. Dilation by a factor of 3
D. Reflection in the origin
E. Trivial transformation (i.e. T(v)=0 for all v)
F. Identity transformation (i.e. T(v)=v for all v)

A) Any (x,y,5) is projected onto (0, 0, 5). What would the inverse transformation of (0, 0, 5) be? Or, is there any (x, y, z) that would be "projected" onto (1, 1, 1)?

B) What happens if you rotate by angle $\displaystyle \theta$ and then rotate by angle $\displaystyle -\theta$?

C) What happens if you "dilate" by a factor of 3 and then "dilate" by a factor of 1/3?

D) What happens if you reflect twice?

E) So any (x, y, z) is tranformed to (0, 0,0). What would the inverse transform of (0, 0, 0) be? What would the inverse transform of (1, 1, 1) be?

F) What happens if you apply this transformation twice?