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Unformatted text preview: Quiz #2
1. Let V and W be ﬁnitedimensional vector spaces (over F ) with ordered
bases α and β, respectively. And let T be a linear transformation from
V to W . Label the following statements as being true or false. No
veriﬁcation is needed.
(a) Given x1 , x2 ∈ V and y1 , y2 ∈ W , there exists a linear transformation U : V → W such that U (x1 ) = y1 and U (x2 ) = y2 .
(Solution) False
(b) [T (v)]β = [T ]β [v]α for all v ∈ V .
α
(Solution) True
(c) T is invertible if and only if T is onetoone and onto.
(Solution) True
(d) Let A and B are matrices. Then AB = I implies that A and B
are invertible.
(Solution) False
(e) Let U be a linear operator on V . Then for any ordered bases γ
and γ for V , [U ]γ is similar to [U ]γ .
(Solution) True
2. Let T : R2 → R3 be deﬁned by T (a1 , a2 ) = (a1 − a2 , a1 , 2a1 + a2 ). Let
α = {(1, 2), (2, 3)} and γ = {(1, 1, 0), (0, 1, 1), (2, 2, 3)}. Compute [T ]γ .
α
(Solution)
7
2
T (1, 2) = (−1, 1, 4) = − (1, 1, 0) + 2(0, 1, 1) + (2, 2, 3),
3
3
11
4
T (2, 3) = (−1, 2, 7) = − (1, 1, 0) + 3(0, 1, 1) + (2, 2, 3).
3
3
Therefore 7
− 3
γ 2
[T ]α = 2
3 1 11 3
3 . 4
3 − 3. Find the change of coordinate matrix that changes β coordinates into
βcoordinates for the following pair of ordered bases β and β for P2 (R).
β = {2x2 − x + 1, x2 + 3x − 2, −x2 + 2x + 1}
β = {9x − 9, x2 + 21x − 2, 3x2 + 5x + 2}
β (Solution) Note that Q = IP2 (R) β is the change of coordinate matrix that changes β coordinates into βcoordinates. Denote IP2 (R) by
I.
I(9x − 9) = 9x − 9
= −2(2x2 − x + 1) + 3(x2 + 3x − 2) − 1(−x2 + 2x + 1),
I(x2 + 21x − 2) = x2 + 21x − 2
= 1(2x2 − x + 1) + 4(x2 + 3x − 2) + 5(−x2 + 2x + 1),
I(3x2 + 5x + 2) = 3x2 + 5x + 2
= 2(2x2 − x + 1) + 1(x2 + 3x − 2) + 2(−x2 + 2x + 1).
Therefore −2 1 2
Q = 3 4 1 .
−1 5 2 2 ...
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This note was uploaded on 10/02/2009 for the course MATH 54554 taught by Professor Holtz during the Spring '06 term at University of California, Berkeley.
 Spring '06
 Holtz
 Linear Algebra, Algebra

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