MTH5112 Linear Algebra I 2012–2013
Coursework 8
Please hand in your solutions of the starred feedback exercises by noon on Friday 30 November
using the red Linear Algebra I Collection Box in the Basement. Don’t forget to put your name
(with your surname underlined) and student number on your solutions, and to staple them.
Exercise 1. Determine which of the following are linear transformations from R3to R2:
(a) L(x) =(x2+x3, x3+x1)T,(b) L(x) =(x1+x2,1)T,
(c) L(x) =(x2−2x1, x3)T,(d) L(x) =(x2
1, x2)T.
Exercise 2. Let Cbe a fixed n×nmatrix. Determine which of the following are linear transfor-
mations from Rn×nto Rn×n:
(a) L(A) = 5A , (b) L(A) = AC +CA ,
(c) L(A) = ACA , (d) L(A) = CAC .
Exercise 3. Determine which of the following are linear transformations from P2to P2:
(a) (L(p))(t) = t2p(0) ,(b) (L(p))(t) = p0(t)−t .
Exercise* 4. Let Mbe a fixed n×nmatrix with M6=O. Determine which of the following are
linear transformations and justify your answers.
(a) L:R2→R3, L(x) = (1 + x1,2 + x2,3)T; (b) L:R3→R2, L(x) = (x3−x2, x2−x1)T;
(c) L:Rn×n→Rn×n, L(A) = AT; (d) L:Rn×n→Rn×n, L(A) = AM2;
(e) L:Rn×n→Rn×n, L(A) = A2M; (f) L:C[0,1] →C[0,1] ,(L(f))(t) = 2t .
Exercise 5. A linear transformation L:V→Wis said to be one-to-one if L(v1) = L(v2)implies
that v1=v2(that is, no distinct vectors v1,v2∈Vget mapped to the same vector w∈W).
Show that Lis one-to-one if and only if ker(L) = {0}.
Exercise 6. For each of the following linear transformations find its matrix representation with
respect to the standard bases of R2and R3:
(a) L:R3→R2where L(x) = (x1−2x3, x1+ 7x2)T;
(b) L:R3→R3where L(x) = (x2+x3,5x1−x3, x2−4x1)T;
(c) L:R2→R3where L(x) = (x2, x1, x1−x2)T.
Exercise* 7. Let L:P2→P2be the mapping given by
(L(p))(t) = p(t−1) ,
and let P={p1,p2,p3}be the standard basis of P2given by
p1(t) = 1 ,p2(t) = t , p3(t) = t2.
(a) Show that Lis a linear transformation.
(b) Find the matrix representation of Lwith respect to P.