Suppose that f : R² → R²is a linear transformation. The figure shows a basis B = {b1, b2} for R- for the domain and codomain (in black), its B-coordinate grid in both the domain and the codomain (in gray), a vector v in the domain (in red), and vectors f(b1) and f(b2) in the codomain (in blue). ty f(b2) b2 b1 f(b1) B-coordinate grid B-coordinate grid • Part 1: Geometric properties of the linear transformation a. Write the vectors f(bj) and f(b2) as linear combinations of the vectors in the basis B. Enter a vector sum of the form 5 b1 + 6 b2. f(b1) = f(b2) = b. The vector b, choose an eigenvector for the linear transformation f with eigenvalue (enter a number or DNE). The vector b, choose an eigenvector for the linear transformation f with eigenvalue (enter a number or DNE).
Suppose that f : R² → R²is a linear transformation. The figure shows a basis B = {b1, b2} for R- for the domain and codomain (in black), its B-coordinate grid in both the domain and the codomain (in gray), a vector v in the domain (in red), and vectors f(b1) and f(b2) in the codomain (in blue). ty f(b2) b2 b1 f(b1) B-coordinate grid B-coordinate grid • Part 1: Geometric properties of the linear transformation a. Write the vectors f(bj) and f(b2) as linear combinations of the vectors in the basis B. Enter a vector sum of the form 5 b1 + 6 b2. f(b1) = f(b2) = b. The vector b, choose an eigenvector for the linear transformation f with eigenvalue (enter a number or DNE). The vector b, choose an eigenvector for the linear transformation f with eigenvalue (enter a number or DNE).
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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Question
![Suppose that f : R² → R² is a linear transformation. The figure shows a basis B = {b1, b2} for R²
for the domain and codomain (in black), its B-coordinate grid in both the domain and the codomain (in
gray), a vector v in the domain (in red), and vectors f(b1) and f(b2) in the codomain (in blue).
ty
f(b2)
b2
[f)B
f(b1)
B-coordinate grid
B-coordinate grid
• Part 1: Geometric properties of the linear transformation
a. Write the vectors f(b¡) and f(b2) as linear combinations of the vectors in the basis B. Enter
a vector sum of the form 5 b1 + 6 b2.
f(b1) :
f(b2)
b. The vector b1
choose
an eigenvector for the linear transformation f
with eigenvalue
(enter a number or DNE).
The vector b choose
an eigenvector for the linear transformation f
with eigenvalue
(enter a number or DNE).
• Part 2: The matrix of the linear transformation relative to the basis B
a. Find the B-coordinate vectors for b, and b2. Enter your answers as coordinate vectors of the
form <5,6>.
[b1]B =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce0afc3b-3e57-4487-b673-cf6f07e8890b%2F1839a848-f855-4428-8725-0d9f12d3547d%2Fp7k5ge_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that f : R² → R² is a linear transformation. The figure shows a basis B = {b1, b2} for R²
for the domain and codomain (in black), its B-coordinate grid in both the domain and the codomain (in
gray), a vector v in the domain (in red), and vectors f(b1) and f(b2) in the codomain (in blue).
ty
f(b2)
b2
[f)B
f(b1)
B-coordinate grid
B-coordinate grid
• Part 1: Geometric properties of the linear transformation
a. Write the vectors f(b¡) and f(b2) as linear combinations of the vectors in the basis B. Enter
a vector sum of the form 5 b1 + 6 b2.
f(b1) :
f(b2)
b. The vector b1
choose
an eigenvector for the linear transformation f
with eigenvalue
(enter a number or DNE).
The vector b choose
an eigenvector for the linear transformation f
with eigenvalue
(enter a number or DNE).
• Part 2: The matrix of the linear transformation relative to the basis B
a. Find the B-coordinate vectors for b, and b2. Enter your answers as coordinate vectors of the
form <5,6>.
[b1]B =
![form <5,6>.
[b1]B =
[b2]B
b. Find the B-coordinate vectors for f(bj) and f(b2). Enter your answers as coordinate vectors
of the form <5,6>.
[f(b1)]B =
[f(b2)]B
c. Find the matrix M for the linear transformation f relative to the basis B in both the domain and
codomain. That is, find the matrix M such that [f(v)]g = M[v]g.
M
• Part 3: Evaluating the linear transformation
a. Write the red vector v in the domain as a linear combination of the vectors in the basis B. Enter
a vector sum of the form 5 b1 + 6 b2.
V =
b. Using the properties of linear transformations, write the vector f(v) in the codomain as a linear
combination of the vectors in the basis B. Enter a vector sum of the form 5 b1 + 6 b2. You
should be able to draw the vector f(v) in the codomain.
f(v)
c. Find the B-coordinate vector for v. Enter your answer as a coordinate vector of the form <5,6>.
[v]B
d. Find the B-coordinate vector for f(v). Enter your answer as a coordinate vector of the form
<5,6>. Hint: use that [f(v)]B = M[v]B.
[f(v)]B](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce0afc3b-3e57-4487-b673-cf6f07e8890b%2F1839a848-f855-4428-8725-0d9f12d3547d%2F0yubh_processed.png&w=3840&q=75)
Transcribed Image Text:form <5,6>.
[b1]B =
[b2]B
b. Find the B-coordinate vectors for f(bj) and f(b2). Enter your answers as coordinate vectors
of the form <5,6>.
[f(b1)]B =
[f(b2)]B
c. Find the matrix M for the linear transformation f relative to the basis B in both the domain and
codomain. That is, find the matrix M such that [f(v)]g = M[v]g.
M
• Part 3: Evaluating the linear transformation
a. Write the red vector v in the domain as a linear combination of the vectors in the basis B. Enter
a vector sum of the form 5 b1 + 6 b2.
V =
b. Using the properties of linear transformations, write the vector f(v) in the codomain as a linear
combination of the vectors in the basis B. Enter a vector sum of the form 5 b1 + 6 b2. You
should be able to draw the vector f(v) in the codomain.
f(v)
c. Find the B-coordinate vector for v. Enter your answer as a coordinate vector of the form <5,6>.
[v]B
d. Find the B-coordinate vector for f(v). Enter your answer as a coordinate vector of the form
<5,6>. Hint: use that [f(v)]B = M[v]B.
[f(v)]B
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