IK, ar U₁ = 2 vecte U2 = U1, U2, U3 3a + 3 2a +3 a²+3a 7 113 = 4a - 2 3а-1 2a² a) Show that the rank of the matrix [u₁ U₂ u3] does not depend on a. Hint: Find the rank by putting the matrix in row-echelon form. b) Making reference to a fact from the course, show that u₁, U2, U3 never span R³, no matter what value a takes. c) Express u3 as a linear combination of u₁ and u2, i.e., as c₁u₁ + c2u2. The scalars c₁ and c₂ will depend on a.
IK, ar U₁ = 2 vecte U2 = U1, U2, U3 3a + 3 2a +3 a²+3a 7 113 = 4a - 2 3а-1 2a² a) Show that the rank of the matrix [u₁ U₂ u3] does not depend on a. Hint: Find the rank by putting the matrix in row-echelon form. b) Making reference to a fact from the course, show that u₁, U2, U3 never span R³, no matter what value a takes. c) Express u3 as a linear combination of u₁ and u2, i.e., as c₁u₁ + c2u2. The scalars c₁ and c₂ will depend on a.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Let a € R, and define vectors u₁, U₂, U3 € R³ in terms of a by
- 8.
a
U₁ =
2 U₂ =
3a + 3
2a +3
a² + 3a
7
[4a - 2]
13 = 3a-1
2a²
(a) Show that the rank of the matrix [u₁ u₂ u3] does not depend on
a. Hint: Find the rank by putting the matrix in row-echelon form.
(b) Making reference to a fact from the course, show that u₁, U₂, U3 never
span R³, no matter what value a takes.
(c) Express u3 as a linear combination of u₁ and u2, i.e., as c₁u₁ + c₂U2.
The scalars c₁ and c₂ will depend on a.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87351664-192b-44e9-8681-fefec31a9dea%2Fbe51630c-add6-47f2-b70f-12ac6e5dcdfe%2Fxm6abh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let a € R, and define vectors u₁, U₂, U3 € R³ in terms of a by
- 8.
a
U₁ =
2 U₂ =
3a + 3
2a +3
a² + 3a
7
[4a - 2]
13 = 3a-1
2a²
(a) Show that the rank of the matrix [u₁ u₂ u3] does not depend on
a. Hint: Find the rank by putting the matrix in row-echelon form.
(b) Making reference to a fact from the course, show that u₁, U₂, U3 never
span R³, no matter what value a takes.
(c) Express u3 as a linear combination of u₁ and u2, i.e., as c₁u₁ + c₂U2.
The scalars c₁ and c₂ will depend on a.
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