Linear algebra If Ua = {(x1,x2, x3)T E R³|x1 + 2x2 + 3ax3 = a} is closed under addition, then a = 0.Prove that the above statement is true, or construct a counter-example to disprove it. Set a = 0 gives U, = {(x1,x2, 13)" E R³[T1 + 2x2 = 0}. Define an operation e for thevectors in R by the rule:ūei = (u1, u2, u3)' + (v1, v2, v3) = (u1 + v1, u2 – V2,0)Suppose ū, i e Uo, show that ū J e U, if and only if ū lies on the z-axis.

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Answered: If Ua = {(x1, x2, x3)" E R³|x1+ 2x2 +…

If Ua = {(x1, x2, x3)" E R³|x1+ 2x2 + 3ax3 = a} is closed under addition, then a = 0. Prove that the above statement is true, or construct a counter-example to disprove it.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.2: Properties Of Division

Problem 51E

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Question

Linear algebra

If Ua = {(x1,x2, x3)T E R³|x1 + 2x2 + 3ax3 = a} is closed under addition, then a = 0.
Prove that the above statement is true, or construct a counter-example to disprove it.

Set a = 0 gives U, = {(x1,x2, 13)" E R³[T1 + 2x2 = 0}. Define an operation e for the
vectors in R by the rule:
ūei = (u1, u2, u3)' + (v1, v2, v3) = (u1 + v1, u2 – V2,0)
Suppose ū, i e Uo, show that ū J e U, if and only if ū lies on the z-axis.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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