Suppose f: R² → R² is a linear transformation. The two pictures on top in the figure use standard S-coordinates, where S = {e₁,e₂}. The two figures on bottom in the figure use B-coordinates, where B = {b₁,b₂}. The figure shows the vectors b₁ and b₂ in blue and the vectors f(b₁) and f(b₂) in red. Standard basis S= {e₁,e₂} y 4 3 2 1 -1 -2 -9 -4 -4-3-2-1 b1 b1 b2 [id] ↑ 19 1 2 3 b2 Custom basis B = {b₁,b₂} 4 X [f]'s [f] B →> 4 3 2 1 -1 -2 -9 -4 Standard basis S = {e₁,e₂} y -4-3-2-1 f(b2) पं b1 b2 19 1 2 [id] ↑ b1 b2 44 (b1) 3 4 Custom basis B= {b₁,b₂} X
Suppose f: R² → R² is a linear transformation. The two pictures on top in the figure use standard S-coordinates, where S = {e₁,e₂}. The two figures on bottom in the figure use B-coordinates, where B = {b₁,b₂}. The figure shows the vectors b₁ and b₂ in blue and the vectors f(b₁) and f(b₂) in red. Standard basis S= {e₁,e₂} y 4 3 2 1 -1 -2 -9 -4 -4-3-2-1 b1 b1 b2 [id] ↑ 19 1 2 3 b2 Custom basis B = {b₁,b₂} 4 X [f]'s [f] B →> 4 3 2 1 -1 -2 -9 -4 Standard basis S = {e₁,e₂} y -4-3-2-1 f(b2) पं b1 b2 19 1 2 [id] ↑ b1 b2 44 (b1) 3 4 Custom basis B= {b₁,b₂} X
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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![Suppose ƒ: R² → R² is a linear transformation. The two pictures on top in the figure use standard S-coordinates, where S = {e₁,e₂}. The two figures on bottom in the figure use B-coordinates, where B = {b₁,b₂}. The
figure shows the vectors b₁ and b₂ in blue and the vectors f(b₁) and f(b₂) in red.
4
3
2
1
-1
-2
-3
-4
Standard basis S = {e₁,e₂}
y
-4-3-2-1
b1
b2
19
1 2 3 4
[id] ↑
b1 b2
Custom basis B = {b₁,b₂}
X
*
[f] B
4
3
2
1
-1
-2
-3
Standard basis S = {e₁,e₂}
y
-4-3-2-1
f(b2)
b1 b2
[id] ↑
14
1 2 3
b1
b2
F(b1)
Custom basis B = {b₁,b₂}
X](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ecf562f-d9e7-407a-a6b2-cc10ba03e32e%2F0522fc73-67f7-48d6-86e4-9229cb439f5b%2Fly4g0a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose ƒ: R² → R² is a linear transformation. The two pictures on top in the figure use standard S-coordinates, where S = {e₁,e₂}. The two figures on bottom in the figure use B-coordinates, where B = {b₁,b₂}. The
figure shows the vectors b₁ and b₂ in blue and the vectors f(b₁) and f(b₂) in red.
4
3
2
1
-1
-2
-3
-4
Standard basis S = {e₁,e₂}
y
-4-3-2-1
b1
b2
19
1 2 3 4
[id] ↑
b1 b2
Custom basis B = {b₁,b₂}
X
*
[f] B
4
3
2
1
-1
-2
-3
Standard basis S = {e₁,e₂}
y
-4-3-2-1
f(b2)
b1 b2
[id] ↑
14
1 2 3
b1
b2
F(b1)
Custom basis B = {b₁,b₂}
X
![a. Find the change of basis matrix from B-coordinates to standard S-coordinates. That is, find the matrix B such that [idg(x) = B[x]B.
B=
b. Is A diagonalizable? choose
D B(-1) D B.
A =
✓ If A is diagonalizable, write it as a product of the matrices named B and D (together with matrix operations). For instance, enter your answer using syntax such as
c. Find the matrix A for the linear transformation f relative to the standard basis S in both the domain and codomain. That is, find the matrix A such that [f(x) = A[x]s
A =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ecf562f-d9e7-407a-a6b2-cc10ba03e32e%2F0522fc73-67f7-48d6-86e4-9229cb439f5b%2F1cajmzp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a. Find the change of basis matrix from B-coordinates to standard S-coordinates. That is, find the matrix B such that [idg(x) = B[x]B.
B=
b. Is A diagonalizable? choose
D B(-1) D B.
A =
✓ If A is diagonalizable, write it as a product of the matrices named B and D (together with matrix operations). For instance, enter your answer using syntax such as
c. Find the matrix A for the linear transformation f relative to the standard basis S in both the domain and codomain. That is, find the matrix A such that [f(x) = A[x]s
A =
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