Let 1₁, 12, 13, 14 € R³, and suppose that 5 1 -3 2 0 1 -4 0 00 [1₁ 12 13 14] → 0 0 (The matrix on the left is the 3×4 matrix with columns U₁, U2, U3, U4.) Express each of u₂ and u4 as a linear combination of u₁ and u3. Let V1, V2, V3, V4, V5 € R5, let A = [v₁ V5], and suppose that the system [Ab] is consistent for every b € R5. Show that V₁,..., V5 are linearly independent. Hint: Interpret the statement "the system [Ab] is consistent for every b & R5" in terms of the number of pivots in a row-echelon form for A.
Let 1₁, 12, 13, 14 € R³, and suppose that 5 1 -3 2 0 1 -4 0 00 [1₁ 12 13 14] → 0 0 (The matrix on the left is the 3×4 matrix with columns U₁, U2, U3, U4.) Express each of u₂ and u4 as a linear combination of u₁ and u3. Let V1, V2, V3, V4, V5 € R5, let A = [v₁ V5], and suppose that the system [Ab] is consistent for every b € R5. Show that V₁,..., V5 are linearly independent. Hint: Interpret the statement "the system [Ab] is consistent for every b & R5" in terms of the number of pivots in a row-echelon form for 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
![(b) Let u₁, U2, U3, U4 € R³, and suppose that
[u₁ աշ U3 U₁] →
1 -3 2 5
0 1 -4
0 00 0
0
(The matrix on the left is the 3×4 matrix with columns U₁, U₂, U3, U4.)
Express each of u₂ and u4 as a linear combination of u₁ and 13.
(c) Let V1, V2, V3, V4, V5 € R5, let A
=
[V₁
V5], and suppose that
the system [Ab] is consistent for every b € R5. Show that V₁,..., V5
are linearly independent. Hint: Interpret the statement "the system
[Ab] is consistent for every b R5" in terms of the number of
pivots in a row-echelon form for A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87351664-192b-44e9-8681-fefec31a9dea%2F894c5deb-33e8-475f-a8cf-89d960056386%2F7ey27cr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Let u₁, U2, U3, U4 € R³, and suppose that
[u₁ աշ U3 U₁] →
1 -3 2 5
0 1 -4
0 00 0
0
(The matrix on the left is the 3×4 matrix with columns U₁, U₂, U3, U4.)
Express each of u₂ and u4 as a linear combination of u₁ and 13.
(c) Let V1, V2, V3, V4, V5 € R5, let A
=
[V₁
V5], and suppose that
the system [Ab] is consistent for every b € R5. Show that V₁,..., V5
are linearly independent. Hint: Interpret the statement "the system
[Ab] is consistent for every b R5" in terms of the number of
pivots in a row-echelon form for A.
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