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 (1,1) to (3,7) and (2,0) to (2,6).

Find the matrix 𝐴 which represents this transformation.

  • A 𝐴 =  1 2 3 4 
  • B 𝐴 =  3 1 4 3 
  • C 𝐴 =  3 2 4 1 
  • D 𝐴 =  2 3 1 4 
  • E 𝐴 =  1 2 3 4 

Where does this transformation map (1,0) and (0,1)?

  • A ( 1 , 0 ) β†’ ( 1 , 3 ) , ( 0 , 1 ) β†’ ( 2 , 4 )
  • B ( 1 , 0 ) β†’ ( 3 , 1 ) , ( 0 , 1 ) β†’ ( 4 , 3 )
  • C ( 1 , 0 ) β†’ ( 3 , 2 ) , ( 0 , 1 ) β†’ ( 4 , 1 )
  • D ( 1 , 0 ) β†’ ( 2 , 3 ) , ( 0 , 1 ) β†’ ( 1 , 4 )
  • E ( 1 , 0 ) β†’ ( 1 , 3 ) , ( 0 , 1 ) β†’ ( 3 , 4 )

Q2:

Suppose the linear transformation 𝐿 sends (1,0) to (βˆ’1,5) and (1,1) to (βˆ’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?

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

Q4:

The determinant of a 2Γ—2 matrix is βˆ’1. What is the area of the image of a unit square under the transformation it represents?

Q5:

A linear transformation maps the points 𝐴, 𝐡, 𝐢, and 𝐷 onto π΄βˆ—, π΅βˆ—, πΆβˆ—, 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.

  • A βˆ’ 6 5
  • B 8 5
  • C 6 5
  • D 9 5
  • E βˆ’ 8 5

Q6:

Find the matrix of the transformation that maps the points 𝐴, 𝐡, and 𝐢, onto π΄βˆ—, π΅βˆ—, and πΆβˆ— as shown.

  • A ⎑ ⎒ ⎒ ⎣ βˆ’ 9 5 1 1 1 5 βˆ’ 3 5 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 9 5 1 1 1 5 3 5 βˆ’ 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ βˆ’ 5 9 1 5 1 1 3 5 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ βˆ’ 9 5 1 1 1 5 3 5 βˆ’ 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ βˆ’ 9 5 βˆ’ 3 5 1 1 5 1 7 1 5 ⎀ βŽ₯ βŽ₯ ⎦

Q7:

What is the matrix 𝑀 that sends points 𝐴, 𝐡, and 𝐢 to 𝐴′, 𝐡′, and 𝐢′ as shown?

  • A 𝑀 =  1 2 3 4 
  • B 𝑀 =  3 1 2 4 
  • C 𝑀 =  1 3 4 1 
  • D 𝑀 =  2 3 4 1 
  • E 𝑀 =  1 1 3 1 

Q8:

The matrix 𝐴 represents a linear transformation which sends the vector 10 to ο”π‘π‘žο . What can you say about the matrix 𝐴?

  • AIts first column is ο”π‘π‘žο .
  • BIts first row is [π‘π‘ž].
  • CIts second column is ο”π‘π‘žο .
  • DIts second row is [π‘π‘ž].

Q9:

Suppose that the matrix 𝐴=ο›π‘Žπ‘π‘π‘‘π‘’π‘“π‘”β„Žπ‘–ο§ represents a transformation that sends the vector 001 to itself and sends every vector in the π‘₯𝑦-plane to a (possibly different) vector in the π‘₯𝑦-plane. What can be said about the entries of 𝐴?

  • A 𝑏 = 1 , β„Ž = 1 , 𝑑 = 1 , 𝑓 = 1 , 𝑒 = 0
  • B 𝑐 = 0 , 𝑓 = 0 , 𝑔 = 0 , β„Ž = 0 , 𝑖 = 1
  • C 𝑐 = 1 , 𝑓 = 1 , 𝑔 = 0 , β„Ž = 0 , 𝑖 = 1
  • D 𝑏 = 0 , β„Ž = 0 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 1
  • E 𝑐 = 1 , 𝑓 = 1 , 𝑔 = 1 , β„Ž = 1 , 𝑖 = 0

Q10:

Suppose that the matrix 𝐴=ο›π‘Žπ‘π‘π‘‘π‘’π‘“π‘”β„Žπ‘–ο§ represents a transformation that sends the vector 010 to itself and sends every vector in the π‘₯𝑧-plane to a (possibly different) vector in the π‘₯𝑧-plane. What can be said about the entries of 𝐴?

  • A 𝑏 = 0 , β„Ž = 1 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 0
  • B 𝑏 = 1 , β„Ž = 1 , 𝑑 = 1 , 𝑓 = 1 , 𝑒 = 0
  • C 𝑏 = 0 , β„Ž = 0 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 1
  • D 𝑐 = 0 , 𝑓 = 0 , 𝑔 = 0 , β„Ž = 0 , 𝑖 = 1
  • E 𝑏 = 0 , β„Ž = 0 , 𝑑 = 0 , 𝑓 = 0 , 𝑒 = 0

Q11:

Consider the vector space of polynomials of degree three at most. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the matrix that represents the linear transformation 𝐿=𝐷+2𝐷+1 with respect to the basis 1,π‘₯,π‘₯,π‘₯.

  • A ⎑ ⎒ ⎒ ⎣ 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 1 1 2 0 0 1 1 6 0 0 1 1 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ 1 1 0 0 0 1 2 0 0 0 1 3 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ 1 2 2 0 0 1 4 6 0 0 1 6 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦

Q12:

Consider the vector space of polynomials of degree three at most. The differentiation operator 𝐷 is a linear transformation on this vector space. Find the matrix that represents the linear transformation 𝐿=𝐷+5𝐷+4 with respect to the basis 1,π‘₯,π‘₯,π‘₯.

  • A ⎑ ⎒ ⎒ ⎣ 1 1 0 0 0 1 2 0 0 0 1 3 0 0 0 1 ⎀ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎣ 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎣ 4 1 0 0 0 4 2 0 0 0 4 3 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎣ 4 1 0 0 0 4 1 0 0 0 0 4 1 5 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎣ 4 5 2 0 0 4 1 0 6 0 0 4 1 5 0 0 0 4 ⎀ βŽ₯ βŽ₯ ⎦

Q13:

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

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

Q14:

Let 𝐿 be a linear transformation of β„οŠ¨ into itself with the property that πΏο€Όο”βˆ’64=22 and 𝐿21βˆ’12=ο”βˆ’33.

Using the fact that 6βˆ’4=βˆ’ο€Όο”βˆ’64, find 𝐿6βˆ’4.

  • A  2 2 
  • B  βˆ’ 2 βˆ’ 2 
  • C  βˆ’ 6 4 
  • D  βˆ’ 2 2 
  • E  6 βˆ’ 4 

Using the fact that ο€Όο”βˆ’128=2ο€Όο”βˆ’64, find πΏο€Όο”βˆ’128.

  • A  1 2 6 
  • B  2 2 
  • C  βˆ’ 4 βˆ’ 4 
  • D  4 4 
  • E  βˆ’ 2 8 βˆ’ 4 4 

Find a vector 𝑣 so that 𝐿(𝑣)=11.

  • A  3 βˆ’ 2 
  • B  βˆ’ 2 1 
  • C  1 βˆ’ 2 
  • D  2 2 
  • E  βˆ’ 3 2 

What is 𝐿(4𝑣+𝑀), where 𝑣=ο”βˆ’64 and 𝑀=21βˆ’12?

  • A  1 5 βˆ’ 8 
  • B  βˆ’ 1 0 1 4 
  • C  5 1 1 
  • D  βˆ’ 4 2 0 
  • E  1 1 2 9 

By considering suitable linear combinations of ο€Όο”βˆ’64 and 21βˆ’12, find 𝐿10 and 𝐿01.

  • A  1 2  ,  1 2 
  • B  3 5  ,  1 2 
  • C  3 5  ,  3 5 
  • D  1 3  ,  2 5 
  • E  1 2  ,  3 5 

Find the matrix 𝑀 which represents the linear transformation 𝐿.

  • A 𝑀 =  1 3 2 5 
  • B 𝑀 =  3 5 1 2 
  • C 𝑀 =  1 2 1 2 
  • D 𝑀 =  3 5 3 5 
  • E 𝑀 =  1 2 3 5 

Q15:

The vertex matrix of a square of side 1 shown is ⎑⎒⎒⎣1212βˆ’12βˆ’1232525232⎀βŽ₯βŽ₯⎦.

Determine the vertex matrix of the image after a transformation by the matrix 1221, and state what geometric figure it is.

  • A ⎑ ⎒ ⎒ ⎣ 5 2 7 2 1 1 2 9 2 3 2 1 2 5 2 1 1 2 ⎀ βŽ₯ βŽ₯ ⎦ , a parallelogram
  • B ⎑ ⎒ ⎒ ⎣ 5 2 7 2 1 1 2 9 2 3 2 1 2 5 2 1 1 2 ⎀ βŽ₯ βŽ₯ ⎦ , a rectangle
  • C ⎑ ⎒ ⎒ ⎣ 7 2 1 1 2 9 2 5 2 5 2 7 2 3 2 1 2 ⎀ βŽ₯ βŽ₯ ⎦ , a rhombus
  • D ⎑ ⎒ ⎒ ⎣ 7 2 1 1 2 9 2 5 2 5 2 7 2 3 2 1 2 ⎀ βŽ₯ βŽ₯ ⎦ , a square

Q16:

Find the matrix of the transformation that maps the points π‘Ž, 𝑏, and 𝑐 onto π‘ŽοŽ˜, π‘οŽ˜, and π‘οŽ˜ as shown in the figure.

  • A ⎑ ⎒ ⎒ ⎣ 3 9 1 1 βˆ’ 2 0 1 1 βˆ’ 1 3 1 1 1 4 1 1 ⎀ βŽ₯ βŽ₯ ⎦
  • B  3 βˆ’ 1 βˆ’ 1 1 
  • C ⎑ ⎒ ⎒ ⎣ 3 9 1 4 βˆ’ 2 9 1 4 βˆ’ 1 3 1 4 1 9 1 4 ⎀ βŽ₯ βŽ₯ ⎦
  • D  3 8 1 1 βˆ’ 1 8 1 1 βˆ’ 1 1 
  • E  3 βˆ’ 1 βˆ’ 1 5 1 3 1 6 1 3 ο₯

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