# Worksheet: Matrix of Linear Transformation

In this worksheet, we will practice finding the matrix of linear transformation and the image of a vector under transformation.

Q1:

Consider the linear transformation which maps to and to .

Find the matrix which represents this transformation.

• A
• B
• C
• D
• E

Where does this transformation map and ?

• A,
• B,
• C,
• D,
• E,

Q2:

Suppose the linear transformation sends to and to . What is the absolute value of the determinant of the matrix representing ?

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?

• AIt preserves angles.
• BIt has determinant 1 or .
• CIt is either a rotation or a reflection.
• DIt preserves distances.

Q4:

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

Q5:

A linear transformation maps the points , , , and onto , , , and , as shown. By finding the areas of the object and image, and considering orientation, find the determinant of the matrix representing this transformation.

• A
• B
• C
• D
• E

Q6:

Find the matrix of the transformation that maps the points , , and , onto , , and as shown. • A
• B
• C
• D
• E

Q7:

What is the matrix that sends points , , and to , , and as shown? • A
• B
• C
• D
• E

Q8:

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

• AIts first column is .
• BIts first row is .
• CIts second column is .
• DIts second row is .

Q9:

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 ?

• A, , , ,
• B, , , ,
• C, , , ,
• D, , , ,
• E, , , ,

Q10:

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 ?

• A, , , ,
• B, , , ,
• C, , , ,
• D, , , ,
• E, , , ,

Q11:

Consider the vector space of polynomials of degree three at most. The differentiation operator is a linear transformation on this vector space. Find the matrix that represents the linear transformation with respect to the basis .

• A
• B
• C
• D
• E

Q12:

Consider the vector space of polynomials of degree three at most. The differentiation operator is a linear transformation on this vector space. Find the matrix that represents the linear transformation with respect to the basis .

• A
• B
• C
• D
• E

Q13:

Let be the transformation produced by a nonzero matrix with a zero determinant. What is the image of a unit square under ?

• Aanother square
• Ba line segment containing the origin
• Ca single point
• Da rhombus
• Ea parallelogram

Q14:

Let be a linear transformation of into itself with the property that and .

Using the fact that , find .

• A
• B
• C
• D
• E

Using the fact that , find .

• A
• B
• C
• D
• E

Find a vector so that .

• A
• B
• C
• D
• E

What is , where and ?

• A
• B
• C
• D
• E

By considering suitable linear combinations of and , find and .

• A,
• B,
• C,
• D,
• E,

Find the matrix which represents the linear transformation .

• A
• B
• C
• D
• E

Q15:

The vertex matrix of a square of side 1 shown is Determine the vertex matrix of the image after a transformation by the matrix , and state what geometric figure it is.

• A, a parallelogram
• B, a rectangle
• C, a rhombus
• D, a square

Q16:

Find the matrix of the transformation that maps the points , , and onto , , and as shown in the figure. • A
• B
• C
• D
• E