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).

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...

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Question

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=U₁U₂, (provide a justification).

Sub-Task 5. Whether R⁴=U₁+U₃, (provide a justification). Whether R⁴=U₁U₃, (provide a justification).

Sub-Task 6. Whether R⁴=U₂+U₃, (provide a justification). Whether R⁴=U₂U₃, (provide a justification).

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