MTH5112 Linear Algebra I 2012–2013
Coursework 5
Please hand in your solutions of the starred feedback exercises by noon on Friday 2 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. Explain what is wrong with the following solution to Exercise 7(a) on Coursework 2:
If (I+A)is invertible then
(I+A)(I+A)−1= (I+A)(I−A+A2) = I−A+A2+A−A2+A3=I+A3=I
⇒I=I.
Exercise 2. Let u,v, and wbe vectors in a vector space. Prove that if u+w=v+wthen
u=v. As in the lectures, clearly indicate which vector space axiom you are using at each step.
Exercise 3. In each case, determine whether His a subspace of R3and give reasons for your
answer:
(a) H1=(r, s, t)T
r, s, t ∈Rand 3r+s−2t= 0 ,
(b) H2=(r+ 1,0, r)T
r∈R,
(c) H3=(r, s, t)T
r, s, t ∈Rand r2+s2+t2≤1.
Exercise 4. Let C1[a, b]denote the subset of those functions in C[a, b]that have a continuous
derivative on [a, b]. Explain why C1[a, b]is a subspace of C[a, b].
Exercise 5. Determine the nullspace of the following matrix
A=
1 3 −2 1
−3−9 6 −2
2 6 −4 1
.
Exercise 6. Let Uand Vbe subspaces of a vector space W. Define their intersection U∩Vand
their sum U+Vby
U∩V={w∈W|wbelongs to both U and V },
U+V={w∈W|w=u+vwhere u∈Uand v∈V}.
Show that U∩Vand U+Vare subspaces of W.
Exercise* 7. Which of the following statements (if any) are true? Justify your answers.
(a) H1=(r, s, t, u)T
r, s, t, u ∈Rand r+s−3t+ 5u= 0 is a subspace of R4.
(b) H2={A∈Rn×n|Ais symmetric }is a subspace of Rn×n.
(c) H3={f∈C[0,1] |f(1) = 1 }is a subspace of C[0,1].
(d) S1=(1,0,2)T,(3,0,4)T,(5,0,6)Tspans R3.
(e) S2={M1, M2}is a spanning set for V={A∈R2×2|Ais diagonal }, where
M1=1 0
0 0, M2=0 0
0 1.
