Consider L: R2 R2 given by → 2x1 L + (+₁) = (₁²+1₂). X2 x2, Consider the two bases S = [e₁,e2] and F = [u₁, u2], where - (1¹). U₁ = and its inverse is (1), We find that: (a) The representation matrix A of L with respect to S is 2 A = ([L(er)]s_ [L(e2)]s) = (ii). 1 (b) The representation matrix B of L with respect to F is 2 B = ([L(ur)]F_ [L(u2)]x) = (1 7¹). 0 (c) The transition matrix T from F to S is U2 = T = ([1₁]s (u₂]s) = (11¹), T-1 = We can verify directly that 1 2 (11) 1 1 2 B=T-¹ AT = (¹₁¹) (²) (1)-67 - = 2 1 1
Consider L: R2 R2 given by → 2x1 L + (+₁) = (₁²+1₂). X2 x2, Consider the two bases S = [e₁,e2] and F = [u₁, u2], where - (1¹). U₁ = and its inverse is (1), We find that: (a) The representation matrix A of L with respect to S is 2 A = ([L(er)]s_ [L(e2)]s) = (ii). 1 (b) The representation matrix B of L with respect to F is 2 B = ([L(ur)]F_ [L(u2)]x) = (1 7¹). 0 (c) The transition matrix T from F to S is U2 = T = ([1₁]s (u₂]s) = (11¹), T-1 = We can verify directly that 1 2 (11) 1 1 2 B=T-¹ AT = (¹₁¹) (²) (1)-67 - = 2 1 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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![(4) Consider L: R2 R2 given by
Consider the two bases S = [e₁, e2] and F = [u₁, u2], where
(¹).
(1¹).
x1
2x1
L
- (^₂) = (2,2²4 ¹₂).
x2
+
U₁ =
We find that:
(a) The representation matrix A of L with respect to S is
2
A = ([L(en)]s_[L(e2)]s) = (ii)
9).
1
(b) The representation matrix B of L with respect to F is
and its inverse is
2
B = [L(U₂)]F)
· ([L(ur)]F_ [L(us)]x) = (1 7¹).
0
U2 =
(c) The transition matrix T from F to S is
T = ([m]s [1₂]s) = (11¹),
B = T-¹ AT =
T-1
We can verify directly that
1/
= 1 (41)
2
2
2
- 1 (1 1) (² 9) ( 1 ) = (²1¹).
2
1
0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0c1f0c2-9b0e-4920-818e-773d2a2d90af%2F98b364e8-af76-463e-8b93-ed373bd739fd%2Fnzreudf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(4) Consider L: R2 R2 given by
Consider the two bases S = [e₁, e2] and F = [u₁, u2], where
(¹).
(1¹).
x1
2x1
L
- (^₂) = (2,2²4 ¹₂).
x2
+
U₁ =
We find that:
(a) The representation matrix A of L with respect to S is
2
A = ([L(en)]s_[L(e2)]s) = (ii)
9).
1
(b) The representation matrix B of L with respect to F is
and its inverse is
2
B = [L(U₂)]F)
· ([L(ur)]F_ [L(us)]x) = (1 7¹).
0
U2 =
(c) The transition matrix T from F to S is
T = ([m]s [1₂]s) = (11¹),
B = T-¹ AT =
T-1
We can verify directly that
1/
= 1 (41)
2
2
2
- 1 (1 1) (² 9) ( 1 ) = (²1¹).
2
1
0
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