Let U1, U2, U3 – be the following subspaces of R* Ui={(a, b, c, d) eR*| 2a - 3b + c + d=0, 0a - 3b - 2c - 2d=0}; U2= {a, b, c, d)eR| a=4b; c=d=0}; U3={a, b, c, d)eR| a-b+c-d=0, 2a - 2b + c + d=0}; Sub-Task 1. Find a basis and the dimension for each Ui, i=1, 2, 3 Sub-Task 2. Whether R4 is the sum of U; and Uk, Hint. You must check 3 cases (i, k)=(1, 2), (i, k)=(1, 3), (i, k)=(2, 3) Sub-Task 3. If for some pair (i, k) R4=U+Uk, check whether the sum is direct.

Let U1, U2, U3 – be the following subspaces of R* Ui={(a, b, c, d) eR*| 2a - 3b + c + d=0, 0a - 3b - 2c - 2d=0}; U2= {a, b, c, d)eR| a=4b; c=d=0}; U3={a, b, c, d)eR| a-b+c-d=0, 2a - 2b + c + d=0}; Sub-Task 1. Find a basis and the dimension for each Ui, i=1, 2, 3 Sub-Task 2. Whether R4 is the sum of U; and Uk, Hint. You must check 3 cases (i, k)=(1, 2), (i, k)=(1, 3), (i, k)=(2, 3) Sub-Task 3. If for some pair (i, k) R4=U+Uk, check whether the sum is direct.

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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Question

Let U1, U2, U3 – be the following subspaces of R4
Ui={(a, b, c, d)eR*| 2a - 3b + c + d=0, 0a - 3b - 2c - 2d=0};
U2= {a, b, c, d)eR| a=4b; c=d=0};
U3={a, b, c, d)eR“| a-b+c-d=0, 2a - 2b + c + d=0};
Sub-Task 1. Find a basis and the dimension for each Ui, i=1, 2, 3
Sub-Task 2. Whether R4 is the sum of U; and Uk,
Hint. You must check 3 cases (i, k)=(1, 2), (i, k)=(1, 3), (i, k)=(2, 3)
Sub-Task 3. If for some pair (i, k) R=U+Uk, check whether the sum is direct.

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