a) Consider the following three vectors in R¹: V₁ = (1,1,0,2), V₂ = (-2,3,1,-4)", V3 = (0,5,1,0) i) Determine if these three vectors are linearly independent. ii) Express the vector v = (3,3,0,6) as a linear combination of the three vectors. Can this be achieved in a unique way? Justify your answer.
a) Consider the following three vectors in R¹: V₁ = (1,1,0,2), V₂ = (-2,3,1,-4)", V3 = (0,5,1,0) i) Determine if these three vectors are linearly independent. ii) Express the vector v = (3,3,0,6) as a linear combination of the three vectors. Can this be achieved in a unique way? Justify your answer.
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
![a) Consider the following three vectors in R¹:
V₁ = (1,1,0,2),
v₂ = (-2,3,1,-4)", V3 = (0,5,1,0)
i) Determine if these three vectors are linearly independent.
ii) Express the vector v = (3,3,0,6) as a linear combination of the three vectors.
Can this be achieved in a unique way? Justify your answer.
b) Consider the system of equations Ax = b, where
1
3
3 21
-B
A = 2 6 9 7 and b =
-1 -3 3
4
Find a parametric description of the solutions to Ax = b. (Hint: Every solution to Ax =
bis of the form x = x₁ + xp, where xp 0 particular solution and
xn a solution to the homogeneous equation Ax = 0).
10 Mar
c) Prove that if λ is an eigenvalue of A with eigenvector v, then 2² is an eigenvalue of
A² with eigenvector v.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25955cce-bc1c-416d-b6e5-f8a3ff8113e5%2F3fea4549-a97a-453e-9cc6-a3b0e46fa366%2Fg03qeza_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a) Consider the following three vectors in R¹:
V₁ = (1,1,0,2),
v₂ = (-2,3,1,-4)", V3 = (0,5,1,0)
i) Determine if these three vectors are linearly independent.
ii) Express the vector v = (3,3,0,6) as a linear combination of the three vectors.
Can this be achieved in a unique way? Justify your answer.
b) Consider the system of equations Ax = b, where
1
3
3 21
-B
A = 2 6 9 7 and b =
-1 -3 3
4
Find a parametric description of the solutions to Ax = b. (Hint: Every solution to Ax =
bis of the form x = x₁ + xp, where xp 0 particular solution and
xn a solution to the homogeneous equation Ax = 0).
10 Mar
c) Prove that if λ is an eigenvalue of A with eigenvector v, then 2² is an eigenvalue of
A² with eigenvector v.
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