EXERCISES FOR THE COURSE MATHEMATICS, CLABE – 17/04/2018
(1) Determine the ~a, ~b ∈ R2 for which the following maps ϕ : R2 → R2 are linear:
(i) ϕ(~x) = (~a · ~x)~a + ~b;
(ii) ϕ(~x) = 2(~a · ~x)~b;
1
~
~
(iii) ϕ(~x) = (~a · b)~x + b −
;
−2
(iv) ϕ(~x) = (~a · ~b)~x + 2(~a · ~x)~b.
(2) In the cases where the map ϕ in exercise (1) is linear for
−1
1
~
~a =
,
b=
,
1
−2
find the matrix A associated to ϕ w.r.t. the canonical basis and the matrix A0 w.r.t.
the basis
e01 = ~a, e02 = ~b .
Find a basis of the image Im ϕ, the kernel Ker ϕ and say if ϕ is invertible.
(3) Let ϕ : R3 → R3 be the map defined by
  

x1
x1 + x3
ϕ x2  = 2x2 + 3x3  .
x3
−x3
Show that ϕ is linear, find a basis of the image Im ϕ, the kernel Ker ϕ and check the
dimension theorem. Find the matrix A associated to ϕ w.r.t. the canonical basis
and the matrix A0 w.r.t. the basis
e01 = e1 + e2 , e02 = e1 , e03 = e1 − e2 + e3 ,
(*)
without using the matrix of basis change. Say if ϕ is invertible and, if it is, find the
matrices associated to ϕ−1 w.r.t. the canonical basis and the basis (*).
(4) Find the matrix A0 of exercise (3) using the matrix A and the matrix of basis change
from the canonical basis to the basis (*).
(5) Let ϕ : R3 → R3 be the map defined by
  

x1
x2
 ,
tx2
ϕ x2  = 
x3
x1 + x2 − x3
where t ∈ R. Show that ϕ is linear, find a basis of the image Im ϕ, the kernel Ker ϕ
and check the dimension theorem. Find the matrix A associated to ϕ w.r.t. the
canonical basis, the matrix A0 w.r.t. the basis (*) and, finally, the matrix A00 w.r.t.
the basis
e001 = e1 − e2 , e002 = −e1 , e003 = 2e1 + e2 + e3
(**)
using the matrix of basis change from (*) to (**). Say if ϕ is invertible.
1