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In this lesson, we will learn how to find the matrix of the linear transformation of reflecting vectors through a given axis.

Q1:

Which of the following are necessary and sufficient conditions on , , , and for the matrix to represent a reflection?

Q2:

Let π be the linear transformation that reflects all vectors in β 2 in the π₯ -axis. Represent π as a matrix and find its eigenvalues and eigenvectors.

Q3:

A vector in is rotated counterclockwise about the origin through an angle of , and the result is reflected in the -axis. Find, with respect to the standard basis, the matrix of this combined transformation.

Q4:

A reflection in a line through the origin sends the vector to . Find the matrix representation of this reflection.

Q5:

Consider the linear transformation which maps a point to its reflection in the -axis.

Find the matrix which represents this transformation.

Where does this transformation map the point ?

Q6:

Suppose π΄ and π΅ are 2 Γ 2 matrices, with π΄ representing a counterclockwise rotation of 3 0 β about the origin and π΅ representing a reflection in the π₯ -axis. What does the matrix π΄ π΅ represent?

Q7:

Q8:

Consider the reflection in the line .

Find the matrix that represents this transformation.

What is the image of the point under this reflection?

Q9:

Suppose π΄ and π΅ are 2 Γ 2 matrices, with π΄ representing a counterclockwise rotation of 3 0 β about the origin and π΅ representing a reflection in the π₯ -axis. What does the matrix π΅ π΄ represent?

Q10:

Let π be the linear transformation that reflects all vectors in β 3 in the π₯ π¦ -plane. Represent π as a matrix and find its eigenvalues and eigenvectors.

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