(Reflections and projections) (a) Let T : R 3 → R 3 be the transformation from the conceptual problems for Chapter 4: T(x) = 1 3   −1 −2 2 −2 2 1 2 1 2   x. Determine the eigenvalues of T, and find a basis for each eigenspace. (b) Remember that T is supposed to be ‘reflection across a plane S’. Explain what the eigenvalues and eigenvectors from (a) mean geometrically. What is their relationship to S? Why does it make sense for the eigenvalues to be 1 and −1?

Step-by-step explanation:

From the given information:

For eigenvalues:

Let not forget that from the question; we are given ; as such eigenvalues will be:

-1,1,1       since

eigenvector for λ = -1

FInd  

we have :

Thus;

eigenvector for λ = -1 ⇒

For λ = 1

Thus ;

As such;

Given that  T is the reflection

Thus for

we must have:

We can now conclude that the reflection pressures length and any other eigenvalue would mean reflection is stretching or shrinking a vector.

a

step-by-step explanation: