Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Please verify your account before proceeding.

In this lesson, we will learn how to find the matrix of linear transformation which rotates every vector by a given angle and reflects them in the x- or y-axis.

Q1:

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in β 2 through an angle of π 3 and then reflects it across the π¦ -axis.

Q2:

A linear transformation is formed by reflecting every vector in β 2 in the π¦ -axis and then rotating the resulting vector through an angle of π 4 . Find the matrix of this linear transformation.

Q3:

A linear transformation is formed by rotating every vector in β 2 through an angle of π 6 , reflecting the resulting vector in the π₯ -axis, and finally reflecting this vector in the π¦ -axis. Find the matrix of this linear transformation.

Q4:

A linear transformation is formed by rotating every vector in β 2 through an angle of 2 π 3 and then reflecting the resulting vector in the π₯ -axis. Find the matrix of this linear transformation.

Q5:

A linear transformation is formed by reflecting every vector in β 2 across the π¦ -axis and then rotating the resulting vector through an angle of π 6 . Find the matrix of this linear transformation.

Q6:

A linear transformation is formed by rotating every vector in β 2 through an angle of π 3 and then reflecting the resulting vector in the π₯ -axis. Find the matrix of this linear transformation.

Q7:

A linear transformation is formed by reflecting every vector in β 2 in the π₯ -axis and then rotating the resulting vector through an angle of π 6 . Find the matrix of this linear transformation.

Q8:

A linear transformation is formed by reflecting every vector in β 2 in the π₯ -axis and then rotating the resulting vector through an angle of π 4 . Find the matrix of this linear transformation.

Q9:

Let the matrix represent rotation in the plane through an angle of and let the matrix represent reflection in the -axis.

What is the matrix ?

Q10:

Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the -axis.

Find .

Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the -axis and the line of reflection.

What, therefore, is the slope of the line of reflection?

If lies in the direction of the line of reflection, what is ?

By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.

Q11:

Consider the given figure.

The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

Using the fact that is perpendicular to , find the equation of .

Using the fact that , find the coordinates of and .

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

Q12:

A linear transformation is formed by rotating every vector in β 3 through an angle of 3 0 β counterclockwise (when viewed from the positive π§ -axis) about the π§ -axis and then reflecting the resulting vector in the π₯ π¦ -plane. Find the matrix of this linear transformation.

Donβt have an account? Sign Up