MTH5112 Linear Algebra I 2012–2013
Coursework 6
Please hand in your solution of the starred feedback exercise by noon on Friday 16 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 collections of vectors in R3are linearly independent:
(a) (1,1,1)T,(3,4,3)T,(2,1,3)T,(1,1,3)T;
(b) (2,−1,5)T,(1,3,2)T,(3,2,7)T;
(c) (3,3,−6)T,(−2,−1,4)T,(1,4,−1)T;
(d) (1,2,3)T,(4,5,0)T.
Exercise 2. Show the following:
(a) If {v1,...,vn}is a spanning set for a vector space Vand vis any vector in V, then
v,v1,...,vnare linearly dependent.
(b) If v1,...,vnare linearly independent vectors in a vector space V, then v2,...,vncannot
span V.
Exercise 3. Let x1,...,xkbe linearly independent vectors in Rn, and let A∈Rn×nbe invertible.
Define yi=Axifor i= 1, . . . , k. Show that y1,...,ykare linearly independent.
Exercise 4. For each of the collection of vectors in Exercise 1, decide whether they form a basis
for R3. Justify your answer.
Exercise 5. Let Hbe the subspace of P3consisting of all polynomials p∈P3with p(1) = 0. Find
a basis for Hand determine its dimension.
Exercise* 6. Which of the following statements (if any) are true? Justify your answers.
(a) (6,5,4)T,(3,2,1)T,(0,−1,−2)T,(−3,−4,−5)Tare linearly independent vectors in R3.
(b) (1,1,2)T,(−2,1,−4)T,(2,3,8)Tare linearly independent vectors in R3.
(c) p1,p2,p3are linearly independent vectors in P2, where
p1(t) = 1 ,p2(t) = 1 + t , p3(t) = 1 + t+t2.
(d) 1 0
0−2,0 3
3 0,1 6
6−2are linearly independent vectors in R2×2.
(e) If a1,...,anare the columns of an invertible n×nmatrix A, then {a1,...,an}is a basis for
Rn.