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
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Chapter2: Second-order Linear Odes
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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
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 =
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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