Linear Algebra, 2020a

Section V. Change of Basis 265
to the basis 〈⃗β 1 ,...,⃗β i ,...,⃗β j ,...,⃗β n 〉 into one with respect to this basis
〈⃗β 1 ,...,⃗β j ,...,⃗β i ,...,⃗β n 〉.
⃗v = c 1 · ⃗β 1 + ···+ c i · ⃗β i + ···+ c j
⃗β j + ···+ c n · ⃗β n
↦→ c 1 · ⃗β 1 + ···+ c j · ⃗β j + ···+ c i · ⃗β i + ···+ c n · ⃗β n = ⃗v
And, a representation with respect to 〈⃗β 1 ,...,⃗β i ,...,⃗β j ,...,⃗β n 〉 changes via
left-multiplication by a row-combination matrix C i,j (k) into a representation
with respect to 〈⃗β 1 ,...,⃗β i − k⃗β j ,...,⃗β j ,...,⃗β n 〉
⃗v = c 1 · ⃗β 1 + ···+ c i · ⃗β i + c j
⃗β j + ···+ c n · ⃗β n
↦→ c 1 · ⃗β 1 + ···+ c i · (⃗β i − k⃗β j )+···+(kc i + c j ) · ⃗β j + ···+ c n · ⃗β n = ⃗v
(the definition of C i,j (k) specifies that i ≠ j and k ≠ 0).
QED
1.6 Corollary A matrix is nonsingular if and only if it represents the identity map
with respect to some pair of bases.
Exercises
̌ 1.7 In R 2 , where
( ( ) 2 −2
D = 〈 , 〉
1)
4
find the change of basis matrices from D to E 2 and from E 2 to D. Multiply the
two.
1.8 Which of these matrices could be used to change bases?
⎛ ⎞
( ) ( ) ( ) 2 3 −1
1 2 0 −1 2 3 −1
(a)
(b)
(c)
(d) ⎝0 1 0 ⎠
3 4 1 −1 0 1 0
4 7 −2
⎛ ⎞
0 2 0
(e) ⎝0 0 6⎠
1 0 0
̌ 1.9 Find the change of basis matrix for B, D ⊆ R 2 .
( 1
(a) B = E 2 , D = 〈⃗e 2 ,⃗e 1 〉 (b) B = E 2 , D = 〈
2)
( ( ( )
1 1 −1
(c) B = 〈 , 〉, D = E 2 (d) B = 〈 ,
2)
4)
1
( 1
, 〉
4)
( 2
2)
〉, D = 〈
( 0
4)
( 1
, 〉
3)
̌ 1.10 Find the change of basis matrix for each B, D ⊆ P 2 .
(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 〉
1.11 For the bases in Exercise 9, find the change of basis matrix in the other direction,
from D to B.
̌ 1.12 Decide if each changes bases on R 2 . To what basis is E 2 changed?