MTH5112 Linear Algebra I 2012–2013
Coursework 9
Please hand in your solutions of the starred feedback exercises by noon on Friday 7 December
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. Let B={b1,b2,b3}be the basis of R3given in Exercise 2 on Coursework 7, that is,
b1= (1,−2,0)T,b2= (0,1,1)T,b3= (−3,6,1)T,
and let L:R3→R3be the linear transformation given by
L(x) = (3x1+ 3x2+ 3x3,−4x1−5x2+ 6x3,6x1+ 3x2−2x3)T.
(a) Find the matrix representation of Lwith respect to the standard basis of R3.
(b) Using (a) and the results from Exercise 4 (b) on Coursework 7, determine the matrix repre-
sentation of Lwith respect to B.
Exercise* 2. Let L:P2→P2be the linear transformation given by
(L(p))(t) = 2tp(0) + p(1 + 2t).
Let P={p1,p2,p3}and Q={q1,q2,q3}be the bases of P2given by
p1(t) = 1 ,p2(t) = t , p3(t) = t2,and q1(t) = 1−t , q2(t) = 1+2t , q3(t) = 3+7t+2t2.
(a) Find the matrix representation of Lwith respect to P.
(b) Find the transition matrix Sfrom Qto Pand the transition matrix from Pto Q.
(c) Using (a) and (b), determine the matrix representation of Lwith respect to Q.
(d) If p∈P2is given by p(t) = c1(1 −t) + c2(1 + 2t) + c3(3 + 7t+ 2t2)for some c1, c2, c3∈R
and nis a positive integer, find Ln(p).
Exercise 3. Prove the Pythagorean Theorem in Rn, that is, show that two vectors xand yin Rn
are orthogonal if and only if
kx+yk2=kxk2+kyk2.
Exercise 4. Let Hbe a subspace of Rn. Show that H⊥is a subspace of Rn.
Exercise* 5. Let Hbe the subspace of R3spanned by the two vectors y= (1,−1,1)Tand
z= (0,1,−3)T.
(a) Find a basis of H⊥. [Hint: H⊥is the nullspace of a 2×3matrix.]
(b) Give a geometric description of Hand H⊥.
Exercise 6. Let v1,...,vrbe vectors in Rnand let H= Span (v1,...,vr). Show that x∈H⊥if
and only if xis orthogonal to each vjfor j= 1, . . . , r.
Exercise 7. Let A∈Rm×nand let xbe in the column space of A. If ATx=0what is x? [Hint:
Use the Fundamental Subspace Theorem.]