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a2 E R³ are subspaces of R³? (No justification necessary.) Which of the following sets of vectors аз |a1 E R³ such that a1 > 0; • U, the set of all a2 аз E R³ such that a1 + 3a2 = a3; • V, the set of all | a2 a3 • W, the set of all E R³ such that a2 = a2 аз a1 • Z, the set of all a2 E R³ such that a1a2 = 0.

a2 E R³ are subspaces of R³? (No justification necessary.) Which of the following sets of vectors аз |a1 E R³ such that a1 > 0; • U, the set of all a2 аз E R³ such that a1 + 3a2 = a3; • V, the set of all | a2 a3 • W, the set of all E R³ such that a2 = a2 аз a1 • Z, the set of all a2 E R³ such that a1a2 = 0.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Only V is a subspace. The equation a1 + 3a2 = az is equivalent to the homogeneous equation a1 + 3a2 – a3 = 0, so in fact V (the set of solutions of this single homogeneous equation) is the null space of the 3 x 1 matrix 1 3 -1. For U,W, and Z, it is possible to find examples showing that they are not closed under addition, not closed under scalar multiplication, or both.
Transcribed Image Text:Only V is a subspace. The equation a1 + 3a2 = az is equivalent to the homogeneous equation a1 + 3a2 – a3 = 0, so in fact V (the set of solutions of this single homogeneous equation) is the null space of the 3 x 1 matrix 1 3 -1. For U,W, and Z, it is possible to find examples showing that they are not closed under addition, not closed under scalar multiplication, or both.
a1 a2 E R³ are subspaces of R³? (No justification necessary.) Which of the following sets of vectors az |a1 • U, the set of all |a2 E R³ such that ai 2 0; аз • V, the set of all E R³ such that a1 + 3a2 = a3; a2 аз a1 E R³ such that a2 = = a²; • W, the set of all a2 a3 • Z, the set of all |a2 E R³ such that a1a2 = 0. аз
Transcribed Image Text:a1 a2 E R³ are subspaces of R³? (No justification necessary.) Which of the following sets of vectors az |a1 • U, the set of all |a2 E R³ such that ai 2 0; аз • V, the set of all E R³ such that a1 + 3a2 = a3; a2 аз a1 E R³ such that a2 = = a²; • W, the set of all a2 a3 • Z, the set of all |a2 E R³ such that a1a2 = 0. аз
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