Let a € R, and define vectors u₁, U2, U3 € R³ in terms of a by E] 2 U₂ = a U₁ = 3a +3 2a +3 a² + 3a] U3 = [4a-2] 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.
Let a € R, and define vectors u₁, U2, U3 € R³ in terms of a by E] 2 U₂ = a U₁ = 3a +3 2a +3 a² + 3a] U3 = [4a-2] 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.
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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![Let a € R, and define vectors u₁, U2, U3 € R³ in terms of a by
3a + 3
[4a-2]
2a +3
3a - 1
a² + 3a]
2a²
U₁ = 2
a
U₂ =
U3 =
(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%2Fe686847d-ae44-472a-b4ee-4fb39f70bc16%2F450d6192-6ec4-4ba4-a373-12e9d8b48788%2Fbvuqwo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let a € R, and define vectors u₁, U2, U3 € R³ in terms of a by
3a + 3
[4a-2]
2a +3
3a - 1
a² + 3a]
2a²
U₁ = 2
a
U₂ =
U3 =
(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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