Linear Algebra

Section V. Change of Basis 241
Exercises
� 1.6 In R 2 ,where
� � � �
2 −2
D = 〈 , 〉
1 4
find the change of basis matrices from D to E2 and from E2 to D. Multiply the
two.
� 1.7 Find the change of basis matrix for B,D ⊆ R 2 . � � � �
1 1
(a) B = E2, D = 〈�e2,�e1〉 (b) B = E2, D = 〈 , 〉
2 4
� � � �
� � � � � � � �
1 1
−1 2
0 1
(c) B = 〈 , 〉, D = E2 (d) B = 〈 , 〉, D = 〈 , 〉
2 4
1 2
4 3
1.8 ForthebasesinExercise7, find the change of basis matrix in the other direction,
from D to B.
� 1.9 Find the change of basis matrix for each B,D ⊆P2.
(a) B = 〈1,x,x 2 〉,D = 〈x 2 , 1,x〉 (b) B = 〈1,x,x 2 〉,D = 〈1, 1+x, 1+x+x 2 〉
(c) B = 〈2, 2x, x 2 〉,D = 〈1+x 2 , 1 − x 2 ,x+ x 2 〉
� 1.10 Decide if each changes bases on R 2 . To what basis is E2 changed?
(a)
� �
5 0
0 4
(b)
� �
2 1
3 1
(c)
� �
−1 4
2 −8
(d)
� �
1 −1
1 1
1.11 Find bases such that this matrix represents the identity map with respect to
those bases. �
3 1
�
4
2 −1 1
0 0 4
1.12 Conside the vector space of real-valued functions with basis 〈sin(x), cos(x)〉.
Show that 〈2sin(x)+cos(x), 3cos(x)〉 is also a basis for this space. Find the change
of basis matrix in each direction.
1.13 Wheredoesthismatrix�
cos(2θ) sin(2θ)
�
sin(2θ) − cos(2θ)
send the standard basis for R 2 ? Any other bases? Hint. Consider the inverse.
� 1.14 What is the change of basis matrix with respect to B,B?
1.15 Prove that a matrix changes bases if and only if it is invertible.
1.16 Finish the proof of Lemma 1.4.
� 1.17 Let H be a n×n nonsingular matrix. What basis of R n does H change to the
standard basis?
� 1.18 (a) In P3 with basis B = 〈1+x, 1 − x, x 2 + x 3 ,x 2 − x 3 〉 we have this repre-
senatation.
RepB(1 − x +3x 2 − x 3 ⎛ ⎞
0
⎜1⎟
)= ⎝
1
⎠
2
B
Find a basis D giving this different representation for the same polynomial.
RepD(1 − x +3x 2 − x 3 ⎛ ⎞
1
⎜0⎟
)= ⎝
2
⎠
0
D