Worksheet: Properties of Linear Transformations

Q1:

Consider the matrix where .

Find .

• A
• B
• C
• D
• E

Find .

• A
• B2
• C
• D1
• E0

By drawing the image of the unit square under the transformation, identify the geometrical transformation this matrix corresponds to.

• Aa rotation by clockwise about the point
• Ba rotation of clockwise about the origin
• Ca reflection in the line
• Da projection onto the line
• Ea reflection in the line

Q2:

Consider the transformation represented by the matrix

What is the image of the square with vertices , , , and under this transformation?

• Aan arrowhead with vertices , and
• Ba square with vertices , , , and
• Can arrowhead with vertices , and
• Da square with vertices , , , and
• Ea kite with vertices , and

What geometric transformation does this matrix represent?

• Aa dilation with scale factor and center the origin
• Ba stretch in the -direction
• Ca stretch in the -direction
• Da dilation with scale factor 3 and center the origin
• Ea rotation about the origin by an angle of

Q3:

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

Q4:

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

Q5:

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