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.

  • A𝐴=1234
  • B𝐴=3143
  • C𝐴=3241
  • D𝐴=2314
  • E𝐴=1234

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

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

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?

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

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:

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.

  • A65
  • B85
  • C65
  • D95
  • E85

Q6:

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

  • A951115351715
  • B951115351715
  • C591511351715
  • D951115351715
  • E95351151715

Q7:

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

  • A𝑀=1234
  • B𝑀=3124
  • C𝑀=1341
  • D𝑀=2341
  • E𝑀=1131

Q8:

The matrix 𝐴 represents a linear transformation which sends the vector 10 to 𝑝𝑞. What can you say about the matrix 𝐴?

  • AIts first column is 𝑝𝑞.
  • BIts first row is [𝑝𝑞].
  • CIts second column is 𝑝𝑞.
  • DIts second row is [𝑝𝑞].

Q9:

Suppose that the matrix 𝐴=𝑎𝑏𝑐𝑑𝑒𝑓𝑔𝑖 represents a transformation that sends the vector 001 to itself and sends every vector in the 𝑥𝑦-plane to a (possibly different) vector in the 𝑥𝑦-plane. What can be said about the entries of 𝐴?

  • A𝑏=1, =1, 𝑑=1, 𝑓=1, 𝑒=0
  • B𝑐=0, 𝑓=0, 𝑔=0, =0, 𝑖=1
  • C𝑐=1, 𝑓=1, 𝑔=0, =0, 𝑖=1
  • D𝑏=0, =0, 𝑑=0, 𝑓=0, 𝑒=1
  • E𝑐=1, 𝑓=1, 𝑔=1, =1, 𝑖=0

Q10:

Suppose that the matrix 𝐴=𝑎𝑏𝑐𝑑𝑒𝑓𝑔𝑖 represents a transformation that sends the vector 010 to itself and sends every vector in the 𝑥𝑧-plane to a (possibly different) vector in the 𝑥𝑧-plane. What can be said about the entries of 𝐴?

  • A𝑏=0, =1, 𝑑=0, 𝑓=0, 𝑒=0
  • B𝑏=1, =1, 𝑑=1, 𝑓=1, 𝑒=0
  • C𝑏=0, =0, 𝑑=0, 𝑓=0, 𝑒=1
  • D𝑐=0, 𝑓=0, 𝑔=0, =0, 𝑖=1
  • E𝑏=0, =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,𝑥,𝑥,𝑥.

  • A1100011000110001
  • B1120011600110001
  • C1000010000100001
  • D1100012000130001
  • E1220014600160001

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,𝑥,𝑥,𝑥.

  • A1100012000130001
  • B4000040000400004
  • C4100042000430004
  • D410004100004150004
  • E452004106004150004

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 into itself with the property that 𝐿64=22 and 𝐿2112=33.

Using the fact that 64=64, find 𝐿64.

  • A22
  • B22
  • C64
  • D22
  • E64

Using the fact that 128=264, find 𝐿128.

  • A126
  • B22
  • C44
  • D44
  • E2844

Find a vector 𝑣 so that 𝐿(𝑣)=11.

  • A32
  • B21
  • C12
  • D22
  • E32

What is 𝐿(4𝑣+𝑤), where 𝑣=64 and 𝑤=2112?

  • A158
  • B1014
  • C511
  • D420
  • E1129

By considering suitable linear combinations of 64 and 2112, find 𝐿10 and 𝐿01.

  • A12, 12
  • B35, 12
  • C35, 35
  • D13, 25
  • E12, 35

Find the matrix 𝑀 which represents the linear transformation 𝐿.

  • A𝑀=1325
  • B𝑀=3512
  • C𝑀=1212
  • D𝑀=3535
  • E𝑀=1235

Q15:

The vertex matrix of a square of side 1 shown is 1212121232525232.

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

  • A527211292321252112, a parallelogram
  • B527211292321252112, a rectangle
  • C72112925252723212, a rhombus
  • D72112925252723212, a square

Q16:

Find the matrix of the transformation that maps the points 𝑎, 𝑏, and 𝑐 onto 𝑎, 𝑏, and 𝑐 as shown in the figure.

  • A3911201113111411
  • B3111
  • C3914291413141914
  • D3811181111
  • E3115131613

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