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.

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.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.CR: Review Exercises

Problem 78CR: Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set...

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

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.

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