1) Let T be the reflection about the line 4x+3y=0 in the euclidean plane. The st
ID: 2882655 • Letter: 1
Question
1) Let T be the reflection about the line 4x+3y=0 in the euclidean plane. The standard matrix A of T is
ii) A has 2 eigenvalues. One of them is?------- and its corresponding eigenspace is?----------------
iii) The other eigenvalue of A is?--------------- and its coresponding eigenspace is?-------------------
7: Problem 11 Previous Problem List Next (1 point) Let T be the reflection about the line 4a 3y 0 in the euclidean plane. The standard matrix A of T is A has 2 eigenvalues. One of them is and its corresponding eigenspace is span The other eigenvalue of A s and its corresponding eigenspace is span Hint: Use geometric reasoning! Note: In order to get credit for this problem all answers must be correct.Explanation / Answer
Solution:
If we remember the matrix for a reflection, we just need a vector in the direction of the line. When x = 3, y = - 4 so use the vector 3i - 4j = lx i + ly j = or (3,-4) what ever your notation is.
and is normalized by 1/(x^2 + y^2) = 1/(9 + 16) = 1/25
The matrix is
T = [x^2 - y^2 2xy] = [-7/25 -24/25]
[2xy y^2 - x^2] [-24/25 7/25]
T is orthogonal (TT' = I) and so all of the eigenvalues (potentially complex) have modulus 1. Consequently, if the eigenvalues are real they are ±1.
For the eigenvalues () we solve det(TI) = 0.
This gives rise to a quadratic equation for
- {(7/25)^2 ^2} (24/25)^2 = 0
-49/625 + ^2 - 576/625 = 0
^2 = 625/625 = 1
= ±1
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