In this problem, we establish that similar matri- ces describe the same linear transformation relative to different bases. Assume that {e1, e2, ..., en} and {f1, f2, ..., f,} are bases for a vector space V and let T : V → V be a linear transformation. Define the n xn matrices A = [ajk] and B = [bik] by T e)Σαγe, . k = 1, 2, ..., n, (7.3.13) i=1 п T (fR) =bikfi, k = 1,2, ..., n. (7.3.14) If we express each of the basis vectors f1, f2, terms of the basis vectors e1, e2,..., en, we have that in fi = Esjiej, i = 1,2, ..., n, (7.3.15) j=1 for appropriate scalars s ji. Thus, the matrix S = [s ji] describes the relationship between the two bases. or equivalently, (2~) п T &)-ΣΣjbp g. (7.3.16) Use (7.3.13) and (7.3.15) to show that, for k = 1, 2, ...n, we have Τα)-ΣΣaσμ1e (7.3.17) i=1
In this problem, we establish that similar matri- ces describe the same linear transformation relative to different bases. Assume that {e1, e2, ..., en} and {f1, f2, ..., f,} are bases for a vector space V and let T : V → V be a linear transformation. Define the n xn matrices A = [ajk] and B = [bik] by T e)Σαγe, . k = 1, 2, ..., n, (7.3.13) i=1 п T (fR) =bikfi, k = 1,2, ..., n. (7.3.14) If we express each of the basis vectors f1, f2, terms of the basis vectors e1, e2,..., en, we have that in fi = Esjiej, i = 1,2, ..., n, (7.3.15) j=1 for appropriate scalars s ji. Thus, the matrix S = [s ji] describes the relationship between the two bases. or equivalently, (2~) п T &)-ΣΣjbp g. (7.3.16) Use (7.3.13) and (7.3.15) to show that, for k = 1, 2, ...n, we have Τα)-ΣΣaσμ1e (7.3.17) i=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![In this problem, we establish that similar matri-
ces describe the same linear transformation relative
to different bases. Assume that {e1, e2, ..., en} and
{f1, f2, ..., f,} are bases for a vector space V and let
T : V → V be a linear transformation. Define the
n xn matrices A = [ajk] and B = [bik] by
T e)Σαγe, .
k = 1, 2, ..., n, (7.3.13)
i=1
п
T (fR) =bikfi, k = 1,2, ..., n. (7.3.14)
If we express each of the basis vectors f1, f2,
terms of the basis vectors e1, e2,..., en, we have that
in
fi = Esjiej, i = 1,2, ..., n,
(7.3.15)
j=1
for appropriate scalars s ji. Thus, the matrix S = [s ji]
describes the relationship between the two bases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0b4b4bb-8aaf-4f9a-aac7-b82bc89c72cf%2F33378ce9-1b3f-4276-b0b1-cf201ad8119c%2Fwxcx3nr.jpeg&w=3840&q=75)
Transcribed Image Text:In this problem, we establish that similar matri-
ces describe the same linear transformation relative
to different bases. Assume that {e1, e2, ..., en} and
{f1, f2, ..., f,} are bases for a vector space V and let
T : V → V be a linear transformation. Define the
n xn matrices A = [ajk] and B = [bik] by
T e)Σαγe, .
k = 1, 2, ..., n, (7.3.13)
i=1
п
T (fR) =bikfi, k = 1,2, ..., n. (7.3.14)
If we express each of the basis vectors f1, f2,
terms of the basis vectors e1, e2,..., en, we have that
in
fi = Esjiej, i = 1,2, ..., n,
(7.3.15)
j=1
for appropriate scalars s ji. Thus, the matrix S = [s ji]
describes the relationship between the two bases.
Transcribed Image Text:or equivalently,
(2~)
п
T &)-ΣΣjbp g.
(7.3.16)
Use (7.3.13) and (7.3.15) to show that, for k =
1, 2, ...n, we have
Τα)-ΣΣaσμ1e
(7.3.17)
i=1
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