, \"ill 47% 15:21 PM Previous Problem List Next (1 point) Let T be the reflectio

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, "ill 47% 15:21 PM Previous Problem List Next (1 point) Let T be the reflection about the line -4x + 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 is its corresponding eigenspace is and spant Hint: Use geometric reasoning! Note: In order to get credit for this problem all answers must be correct. Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts remaining.

Explanation / Answer

The standard matrix of the linear transformation for reflection across the line y = mx is

(1-m2)/(1+m2)

2m/(1+m2)

2m/(1+m2)

(m2-1)/ (1+m2)

Here, the given line is -4x+3y = 0 or, y = 4x/3 so that m = 4/3. Then the standard matrix of T is A=

-7/25

6/5

6/5

7/25

The eigenvalues of A are solutions to the equation det(A-I2)= 0 or 2 -949/625 = 0 . Thus, the eigenvalues of A are 1 = -949/25 and 2 =949/25.

The eigenvector of A associated with the eigenvalue 1 = -949/25 is solution to the equation (A+949/25 I2)X = 0 which is v1 = (1/30(-7-949),1)T. The related eigenspace is span{(1/30(-7-949),1)T }. Similarly, the eigenvector of A associated with the eigenvalue 2 = 949/25 is solution to the equation (A-949/25 I2)X = 0 which is v2 = (1/30(-7+ 949),1)T. The related eigenspace is span{(1/30(-7+949),1)T }.

(1-m2)/(1+m2)

2m/(1+m2)

2m/(1+m2)

(m2-1)/ (1+m2)

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