369 HW5 problems
1. Let
     
1
0 
 2
β = 2 , 0 , 1 ,


2
1
0
 
4
v = 3 .
4
Show that v is in the span of β. Why doesn’t it make sense to write coordinates of v with respect to β?
2. Let L : R2 → R3 be defined by


x + 2y
x
L
= 2x − 3y  ,
y
y
and let
v=
2
,
−3
1
0
D=
,
,
1
1
     
0
0 
 1
R = 1 , 1 , 0 .


0
1
1
Compute L(v) and [L(v)]R . Then compute [L]RD and [v]D , and check that [L(v)]R = [L]RD [v]D .
3. Let

1
A = 0
3
2
−1
2

−4
1 ,
10
     
0
0 
 1
β = 0 , 1 , 0 .


1
1
1
Show that β is a basis for R3 . Compute [A]std β , [A]β std and [A]ββ .
4. With β and A as in Problem 3, compute also [I]β, std and [I]std, β . Check that [I]−1
β, std = [I]std, β .
Then check that
A = [I]std, β [A]β,β [I]β, std .
5. Let
A=
1
2
2
.
1
Find a basis β = {v1 , v2 } for R2 such that [A]ββ is diagonal. Then find P so that A = P CP −1 , where
C := [A]ββ .
6. Let
A=
1
0
2
,
1
B=
1
1
2
.
2
(a) Is it possible to find a basis β = {v1 , v2 } for R2 such that v1 , v2 have the property Avj = λj vj for
some scalars λj , j = 1, 2? Why or why not?
(b) Find a basis β = {v1 , v2 } for R2 such that v1 , v2 have the property Bvj = λj vj for some scalars
λj , j = 1, 2. Use this to write down [B]ββ . What do the diagonal entries of [B]ββ tell you about the
dimension of the image of B?
1