We consider the vector space R3 which has the standard basis S = {e1,€2, €3}, where e; is the i-th column of the identity matrix I3. 1. Consider the set -{{} [}] } T = 1 (a) Explain why T is a basis of R³. Be sure to fully justify any claims/assertions that you make. (b) Find the matrix A which transforms T into the standard basis S. (In the textbook, this matrix is denoted P_) SET (c) Find the matrix B which transforms the standard basis into T. (In the textbook, this matrix is denoted _P .) TES
We consider the vector space R3 which has the standard basis S = {e1,€2, €3}, where e; is the i-th column of the identity matrix I3. 1. Consider the set -{{} [}] } T = 1 (a) Explain why T is a basis of R³. Be sure to fully justify any claims/assertions that you make. (b) Find the matrix A which transforms T into the standard basis S. (In the textbook, this matrix is denoted P_) SET (c) Find the matrix B which transforms the standard basis into T. (In the textbook, this matrix is denoted _P .) TES
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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![3al 34% 15:19
NOTES ON CHANGE OF BASIS AND LINEAR TRANSFORMATIONS
MICHAEL WOODBURY
Suppose that V is a vector space of dimension n and that each of the sets
S = {e1,€2, . .. , en},
T = {v1, V2, ..., Vm}
U = {w1, W2, ..., wn}
is a basis of V.
11. CHANGE OF BASIS MATRIX
In certain instances, for example if V = R" and S is the standard basis (i.e., e; is the i-th
column of Im), it is very easy to see how to write any element of T or U in terms of S. This
means that finding the coordinate vectors
(1)
[vi]s• [va]g•….. [v»]s
and/or
(2)
is easy. What is harder is to find how to write T in terms of U of vice versa, or more
generally, how to compute [x]T from [x]s or vice versa. Our goal is to come up with a matrix
P, called the change of basis matrir which will do exactly that.
1.1. An example. Let V = P, = {a+ bt | a,b are scalars}, and let
S = {1,t}, T= {1+2, -6+8t}, u = {L+t, L-t}.
w2
Then the coordinate vectors of v, and vz with respect to S are
[vi]s =
and (v:]s =
Note that
[e]s =|
and (e)s =||:
which implies that if
A =
is the matrix whose columns are [v1]s and [v1]s then
Afe) = [: ][:]=[:]=(wila
-6
Date: Spring 2022.
and
In other words, we can think of A as being a transformation that sends the vectors T into
the vectors S. To help us remember that this is what is happening, we write
(3)
- A-
This matrix is called the change of basis matrir from S to T.
Similarly, the coordinate vectors of w1 and wz with respect to S are
and (ws), - [].
%3D
and so the matrix B which has [w,], and [w,], as its columns has the property that it
sends the basis vectors U to S. In other words, the change of basis matrix from U to S is
11](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe32d79fc-6337-43b5-a1cd-895c62f79194%2Ff90ff2a7-abe6-4bb7-9747-fbd87cde74ee%2Fujsj38_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3al 34% 15:19
NOTES ON CHANGE OF BASIS AND LINEAR TRANSFORMATIONS
MICHAEL WOODBURY
Suppose that V is a vector space of dimension n and that each of the sets
S = {e1,€2, . .. , en},
T = {v1, V2, ..., Vm}
U = {w1, W2, ..., wn}
is a basis of V.
11. CHANGE OF BASIS MATRIX
In certain instances, for example if V = R" and S is the standard basis (i.e., e; is the i-th
column of Im), it is very easy to see how to write any element of T or U in terms of S. This
means that finding the coordinate vectors
(1)
[vi]s• [va]g•….. [v»]s
and/or
(2)
is easy. What is harder is to find how to write T in terms of U of vice versa, or more
generally, how to compute [x]T from [x]s or vice versa. Our goal is to come up with a matrix
P, called the change of basis matrir which will do exactly that.
1.1. An example. Let V = P, = {a+ bt | a,b are scalars}, and let
S = {1,t}, T= {1+2, -6+8t}, u = {L+t, L-t}.
w2
Then the coordinate vectors of v, and vz with respect to S are
[vi]s =
and (v:]s =
Note that
[e]s =|
and (e)s =||:
which implies that if
A =
is the matrix whose columns are [v1]s and [v1]s then
Afe) = [: ][:]=[:]=(wila
-6
Date: Spring 2022.
and
In other words, we can think of A as being a transformation that sends the vectors T into
the vectors S. To help us remember that this is what is happening, we write
(3)
- A-
This matrix is called the change of basis matrir from S to T.
Similarly, the coordinate vectors of w1 and wz with respect to S are
and (ws), - [].
%3D
and so the matrix B which has [w,], and [w,], as its columns has the property that it
sends the basis vectors U to S. In other words, the change of basis matrix from U to S is
11
Transcribed Image Text:We consider the vector space R3 which has the standard basis
S = {e1, e2, e3},
%D
where e; is the i-th column of the identity matrix I3.
1. Consider the set
-{{} [}| }
T =
1
(a) Explain why T is a basis of R³. Be sure to fully justify any claims/assertions that
you make.
(b) Find the matrix A which transforms T into the standard basis S. (In the textbook,
this matrix is denoted_P_)
SET
(c) Find the matrix B which transforms the standard basis into T. (In the textbook,
this matrix is denoted_P .)
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