linear transformation

Question

A mapping ?? : R3 → R3 is defined by ??(??1,??2,??3)=(??1 +??2,??2 +??3,??3 +??1). (i) Show that ?? is a linear transformation. (ii) Write down the standard matrix of ?? with respect to the standard basis of R3.  (iii) Show that ?? is a one-to-one transformation.

Answer 1

(i) We show that T( u + v ) = T( u ) + T( v ), where u = <u₁,u₂,u₃>, and v = <v₁,v₂,v₃>, and that T( αu ) = αT( u ), for any scalar α∈R.

 

T( u + v ) = T( u₁ + v₁, u₂ + v₂, u₃ + v₃ )

=  <u₁ + u₂ + v₁ + v₂, u₂ + u₃ + v₂ + v₃, u₃ + u₁ + v₃ + v_1> 

=  T( u ) + T( v ).

T( αu ) = <α(u₁+u₂),α(u₂+u₃),α(u₃+u₁)>

= αT( u ).

 

(ii)

| a  b  c |   | u₁ |     | au₁ + bu₂ + cu₃ |

| d  e  f | * | u₂ | = | du₁ + eu₂ + fu₃  |

| g  h  i |    | u₃ |    |  gu₁ + hu₂ + iu₃ |

 

    | u₁ + u₂ |

=  | u₂ + u₃ |

    | u₃ + u_{1 |}

 

Using the vectors <1,0,0>,<0,1,0>,<0,0,1> for u gives

 a = b = 1, c = 0;

 d = 0, e = f = 1; 

 g = 1, h = 0, i = 1.

 

| 1 1 0 |

| 0 1 1 |

| 1 0 1 |

 

(iii) Verify that the determinant is non-zero, because this implies that the function is one-to-one (and onto): the columns are linearly independent

 

| 1 1 |    | 0 1 |    | 0 1  |

| 0 1 |  - | 1 1 | + | 1 0 |

= 1 + 1 - 1

= 1