Let U1, U2, U3 – be the following subspaces of R4 U1={a, b, c, d) R4| 1b=с; a=d=0}; U2={(a, b, c, d) R4| a - b + 2c + 3d=0, -a +b - 2c - 3d=0}; U3= {a, b, c, d) R4| 1a – 2b – 1c - d=0, a + 2b + 0c + 2 d=0 }; Sub-Task 1. Find a basis and the dimension of U1. Sub-Task 2. Find a basis and the dimension of U2. Sub-Task 3. Find a basis and the dimension of U3. Sub-Task 4. Whether R4=U1+U2, (provide a justification). Whether R4=U1 U2, (provide a justification). Sub-Task 5. Whether R4=U1+U3, (provide a justification). Whether R4=U1 U3, (provide a justification). Sub-Task 6. Whether R4=U2+U3, (provide a justification). Whether R4=U2 U3, (provide a justification).
Let U1, U2, U3 – be the following subspaces of R4 U1={a, b, c, d) R4| 1b=с; a=d=0}; U2={(a, b, c, d) R4| a - b + 2c + 3d=0, -a +b - 2c - 3d=0}; U3= {a, b, c, d) R4| 1a – 2b – 1c - d=0, a + 2b + 0c + 2 d=0 }; Sub-Task 1. Find a basis and the dimension of U1. Sub-Task 2. Find a basis and the dimension of U2. Sub-Task 3. Find a basis and the dimension of U3. Sub-Task 4. Whether R4=U1+U2, (provide a justification). Whether R4=U1 U2, (provide a justification). Sub-Task 5. Whether R4=U1+U3, (provide a justification). Whether R4=U1 U3, (provide a justification). Sub-Task 6. Whether R4=U2+U3, (provide a justification). Whether R4=U2 U3, (provide a justification).
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 U1, U2, U3 – be the following subspaces of R4
U1={a, b, c, d) R4| 1b=с; a=d=0};
U2={(a, b, c, d) R4| a - b + 2c + 3d=0, -a +b - 2c - 3d=0};
U3= {a, b, c, d) R4| 1a – 2b – 1c - d=0, a + 2b + 0c + 2 d=0 };
Sub-Task 1. Find a basis and the dimension of U1.
Sub-Task 2. Find a basis and the dimension of U2.
Sub-Task 3. Find a basis and the dimension of U3.
Sub-Task 4. Whether R4=U1+U2, (provide a justification). Whether R4=U1 U2, (provide a justification).
Sub-Task 5. Whether R4=U1+U3, (provide a justification). Whether R4=U1 U3, (provide a justification).
Sub-Task 6. Whether R4=U2+U3, (provide a justification). Whether R4=U2 U3, (provide a justification).
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