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

Identity transformation

Rotation about the x-axis

Rotation about the y-axis by π will be the correct answer

What is invertible linear transformation ?

Let W and V both have the same finite dimension and be vector spaces over the field F. A linear transformation, T:V→W, shall exist.

If S(T(x))=x for all x∈V, then T is said to be invertible by a linear transformation S:W→V. The opposite of T is known as S. S, to put it simply, reverses all changes T makes to an input x.

In fact, based on the presumptions made at the outset, T is only invertible if it is bijective. Here, we provide evidence that bijectivity necessitates invertibility. As an exercise, try going the other way.

Assume T is a bijective. Consequently, there exists a single vector in W called yx such that T(x)=yx for each x∈V. S:W→V is defined as follows: for every y∈W, Give x∈V a value such that y=yx. The choice is distinct because T is injective, and such an x occurs since T is surjective. As a result, S(T(x))=x and S is a well-defined function from W to V. S still needs to be demonstrated to be a linear transformation.

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