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In this lesson, we will learn how to find the matrix of a linear transformation and how to describe it geometrically.

Q1:

Consider the linear transformation which maps to and to .

Find the matrix which represents this transformation.

Where does this transformation map and ?

Q2:

Suppose the linear transformation 𝐿 sends ( 1 , 0 ) to ( − 1 , 5 ) and ( 1 , 1 ) to ( − 6 , 6 ) . What is the absolute value of the determinant of the matrix representing 𝐿 ?

Q3:

A square of area 1 undergoes a linear transformation. Given that the area of the image is also 1, what can you say about the matrix of the transformation?

Q4:

The determinant of a 2 × 2 matrix is − 1 . What is the area of the image of a unit square under the transformation it represents?

Q5:

Consider the linear transformations for which , the image of , and , the image of , are unit vectors. Let be a linear transformation of this kind which has the additional property that the area of the parallelogram with vertices , , , and is as big as possible. What are the possible values of the size of the angle between and for the transformation ?

Q6:

A linear transformation maps the points 𝐴 , 𝐵 , 𝐶 , and 𝐷 onto 𝐴 ∗ , 𝐵 ∗ , 𝐶 ∗ , and 𝐷 , as shown.

By finding the areas of the object and image, and considering orientation, find the determinant of the matrix representing this transformation.

Q7:

Find the matrix of the transformation that maps the points , , , and onto , , , and as shown.

Q8:

What is the matrix 𝑀 that sends points 𝐴 , 𝐵 , and 𝐶 to 𝐴 ∗ , 𝐵 ∗ , and 𝐶 ∗ as shown?

Q9:

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

Q10:

A linear transformation of a plane sends vector to . If the transformation is a rotation, where does it send ?

Q11:

The matrix represents a linear transformation which sends the vector to . What can you say about the matrix ?

Q12:

Suppose that the matrix represents a transformation that sends the vector to itself and sends every vector in the -plane to a (possibly different) vector in the -plane. What can be said about the entries of ?

Q13:

Q14:

In two-dimensional space you only need to specify the angle of a rotation. In three-dimensional space you need to give both an angle and a vector which represents the axis of rotation. Consider the matrix which represents a rotation of by about an axis through the origin and in the direction of

What is ?

Find the general form of the matrix which sends to the appropriate vector as determined in the previous part of the question.

The vector is perpendicular to . What can you say about the direction of the vector ?

What can you say about the magnitude of ?

Which of the following vectors has the required properties to be ?

What can you say about the vector ?

Using the general form of the matrix from the second part of the question, and the values of and , find the matrix .

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