Linear Algebra, 2020a

Section V. Change of Basis 267
̌ 1.23 Show that the columns of an n×n change of basis matrix form a basis for
R n . Do all bases appear in that way: can the vectors from any R n basis make the
columns of a change of basis matrix?
̌ 1.24 Find a matrix having this effect.
( ( 1 4
↦→
3)
−1)
That is, find a M that left-multiplies the starting vector to yield the ending vector.
Is there
(
a matrix
( )
having
(
these
) (
two
)
effects?
( ( ) ( ( )
1 1 2 −1 1 1 2 −1
(a) ↦→
↦→ (b) ↦→
↦→
3)
1 −1 −1 3)
1 6)
−1
Give a necessary and sufficient condition for there to be a matrix such that ⃗v 1 ↦→ ⃗w 1
and ⃗v 2 ↦→ ⃗w 2 .
V.2 Changing Map Representations
The first subsection shows how to convert the representation of a vector with
respect to one basis to the representation of that same vector with respect to
another basis. We next convert the representation of a map with respect to one
pair of bases to the representation with respect to a different pair — we convert
from Rep B,D (h) to RepˆB, ˆD
(h). Here is the arrow diagram.
V wrt B
id
⏐
↓
V wrt ˆB
h
−−−−→
H
h
−−−−→
Ĥ
W wrt D
id
⏐
↓
W wrt ˆD
To move from the lower-left to the lower-right we can either go straight over,
or else up to V B then over to W D and then down. So we can calculate Ĥ =
RepˆB, ˆD (h) either by directly using ˆB and ˆD, or else by first changing bases with
RepˆB,B (id) then multiplying by H = Rep B,D(h) and then changing bases with
Rep D, ˆD (id).
2.1 Theorem To convert from the matrix H representing a map h with respect
to B, D to the matrix Ĥ representing it with respect to ˆB, ˆD use this formula.
Ĥ = Rep D, ˆD
(id) · H · RepˆB,B
(id)
(∗)
Proof This is evident from the diagram.
QED