Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 15E from Chapter 4.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 15E
Chapter:
Problem:
Let W be the set of all vectors of the form shown, where a, b, and c represent arbitrary real numbers. In each case, either find a set S of vectors that spans W or
Step-by-Step Solution
Given Information
We are given that a set W of the form\[\left[ {\begin{array}{*{20}{c}}{3a + b}\\4\\{a - 5b}\end{array}} \right]\]We have to find whether the given set is a vector space.
Step-1: Check for zero vector
As, one of the element in the vector is 4, the set can not contain zero vector. So W is not a vector space.
Also, the given vector cannot be written as a linear combination of some vectors, because the 4 is not a coefficient of the constants a or bTherefore,
W is not a vector Space
We are given that a set W of the form\[\left[ {\begin{array}{*{20}{c}}{3a + b}\\4\\{a - 5b}\end{array}} \right]\]We have to find whether the given set is a vector space.
Step-1: Check for zero vector
As, one of the element in the vector is 4, the set can not contain zero vector. So W is not a vector space.
Also, the given vector cannot be written as a linear combination of some vectors, because the 4 is not a coefficient of the constants a or bTherefore,
W is not a vector Space