Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 06 (b)
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Question B 06 (b) 

The matrix A below represents a transformation that reflects vectors in in a plane that contains the origin. Note: you can use geometric insight to greatly simplify the calculations below. (b) Find an equation for (equation form). 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

How does the given matrix transform the vectors on the plane? 
Hint 2 

(For alternative solution) draw the columns of this matrix as vectors in . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since the transform reflects vectors with respect to the plane , each vector on the plane is remained as it is, after applying the transformation. In other words, if a vector is in , then . This means that the eigenspace of the eigenvalue is the plane which we are looking for. To find the eigenvectors corresponding to eigenvalue , we solve the equation
This solution to this equation satisfies , so that it can be written as an parameter equation form for the parameters and ;
Therefore, two linearly independent eigenvectors for the eigenvalue are and , and hence the corresponding eigenspace has its normal vector
Using a vector on the plane , we have the equation for the plane as
Answer: 
Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. (Alternative solution) Let be the matrix in this question, and consider the standard basis vectors for
Their images under this transformation are the first, second, and third columns of respectively. Because this transformation is the reflection of about the plane , the points
all lie in as the norm of each of the columns of is equal to the norm of each of the standard basis vectors. The equation of is of the form
where is any vector orthogonal to all vectors in . To find such a vector, we calculate two nonzero points in :
Hence, and are in . So to find a vector orthogonal to all vectors in , solve the following system of equations
To that end, we apply Gaussian elimination to
to obtain
in other words, the set of solutions is given by and . In particular, we can pick as a solution. Hence, the equation of the plane is
