2. Let Q = {[] where x ≤ 0, y ≥ 0} a. Find one vector that is in Q. N Show that Q with the standard addition and multiplication operations, is not a vector space. Find all the axioms that fail to hold. Justify your answer.
2. Let Q = {[] where x ≤ 0, y ≥ 0} a. Find one vector that is in Q. N Show that Q with the standard addition and multiplication operations, is not a vector space. Find all the axioms that fail to hold. Justify your answer.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.1: Vector Spaces And Subspaces
Problem 7EQ
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![Recall the axioms of a vector space: The set V, together with the operations of
addition and scalar multiplication, is said to form a vector space if the following
axioms are satisfied:
i. x+y = y + x for any x and y in V.
ii. (x + y) + z = x + (y+z) for any x, y and 'z in V.
iii. There exists an element O in V such that x + 0 = x for each x E V.
iv. For each x E V, there exists an element -x in Vsuch that x + (-x) = 0.
v. a (x + y) = ax + ay for each scalar a and any x and y in V.
vi. (a + B)x= ax + Bx for each scalar a and ß and any x E V.
vii. (aß)x= a(Bx) for each scalar a and ß and any x E V.
viii. 1x = x for all x E V.
2. Let Q = {[] where x ≤ 0, y ≥ 0}
a.
Find one vector that is in Q.
2
b. Show that Q with the standard addition and multiplication operations, is not a
vector space. Find all the axioms that fail to hold. Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b19024a-5609-4e35-a0a7-7058d3f26c57%2F5b2e3aa6-0863-43ed-8344-b2a45a7579f2%2Fs99oe3.jpeg&w=3840&q=75)
Transcribed Image Text:Recall the axioms of a vector space: The set V, together with the operations of
addition and scalar multiplication, is said to form a vector space if the following
axioms are satisfied:
i. x+y = y + x for any x and y in V.
ii. (x + y) + z = x + (y+z) for any x, y and 'z in V.
iii. There exists an element O in V such that x + 0 = x for each x E V.
iv. For each x E V, there exists an element -x in Vsuch that x + (-x) = 0.
v. a (x + y) = ax + ay for each scalar a and any x and y in V.
vi. (a + B)x= ax + Bx for each scalar a and ß and any x E V.
vii. (aß)x= a(Bx) for each scalar a and ß and any x E V.
viii. 1x = x for all x E V.
2. Let Q = {[] where x ≤ 0, y ≥ 0}
a.
Find one vector that is in Q.
2
b. Show that Q with the standard addition and multiplication operations, is not a
vector space. Find all the axioms that fail to hold. Justify your answer.
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