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Linear Algebra Quiz 3
June 13, 2006 1.
2
1
3
2
,
} and γ = {
,
}.
1
1
1
1
(a)(4 points) Find the transition matrices from β coordinates to standard coordinates, γ coordinates to standard coordinates, and γ coordinates to β coordinates, i.e., ﬁnd [id]STD ,
β
[id]STD , and [id]γ .
γ
β
x
2x + y
(b) (4 points) Let L : R2 → R2 by L
=
. Show that L is a linear transfory
−x + 3y
mation and ﬁnd a matrix A such that L(x) = Ax for all x ∈ R2 .
(c) (4 points) Find the matrix representing L with respect to the bases β to β and β to γ ,
i.e., ﬁnd [L]β and [L]γ .
β
β
3
2
(d) (4 points) Suppose x = 2
−
. Find the coordinate representation of x with
1
1
respect to the ordered bases β , γ , and the standard basis, i.e., ﬁnd [x]β , [x]γ , and [x]STD .
(e) (4 points) Find the coordinate representation of L(x) with respect to the ordered bases
β , γ , and the standard basis, i.e., ﬁnd [L(x)]β , [L(x)]γ , and [L(x)]STD . Consider R2 with ordered bases β = { Solution:
(a) [id]STD =
β
0 −1
.
12
x
(b) L
=
y 21
, [id]STD =
γ
11 21
−1 3 x
y 32
, [id]γ = [id]γ [id]STD =
β
STD
β
11 1 −2
−1 3 21
11 = so L is left multiplication by a matrix and is hence linear. 21
1 −1
so [L]β = [id]β [L]STD [id]STD =
STD
β
STD
β
−1 3
−1 2
41
1 −2
21
2
. Similarly, [L]γ = [id]γ [L]STD [id]STD =
STD
β
STD
β
−3 1
−1 3
−1 3
1
0 −1
2
1
2
3
(d) [x]β =
, [x]STD =
, [x]γ = [id]γ [x]β =
=
β
12
−1
−1
−1
1
41
2
7
(e) [L(x)]β = [L]β [x]β =
=
, [L(x)]γ = [id]γ [L(x)]β =
β
β
−3 1
−1
−7
21
7
7
7
(weird!), [L(x)]STD = [id]STD [L(x)]β =
=
β
−7
11
−7
0 (c) (b) says [L]STD =
STD 1 21
21
=
−1 3
11
1
3 −1
=
1
−2 3
.
0 −1
12 7
−7 = ...
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This note was uploaded on 10/18/2011 for the course S 2010D taught by Professor Peters during the Spring '10 term at Columbia.
 Spring '10
 Peters

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