Inverse of Linear Transformations

For each of the following linear transformations, find the inverse if it exists, or explain why there is no inverse.

(a) T : R 3 → R 3 where T(v) is the reflection of v around the plane x + 2y + 3z = 0.

(b) T : R 3 → R 3 where T(v) is the projection of v along the vector (1, 2, 3)

(c) T : R 3 → R 3 where T(v) = Av and

A =

\begin{matrix} 1 & 1 & b \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{matrix} where b is a real number.

So for these would I just have to put the plane or vector into matrix form and try to find the inverse? Thanks!

Solution 1:

B does not have an inverse. The projective is not injective which can be seen since $\dim (\mathbb{R^3}) = 3$ and $dim(span(1,2,3)) = 1$.

For C just find the inverse of the matrix.

For A find the matrix representation. To do this, take a look at what the transformation does to $(1,2,3)$. Then find two vectors that lie in that plane that are orthogonal to each other. This will give you a basis for $\mathbb{R^3}$ and should help you find the matrix representation.