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If U and V are vector spaces, define the Cartesian product of U and V to be   U X V = {(u, v) : u is in U and vis in V} Prove that U X V is a vector space.

If U and V are vector spaces, define the Cartesian product of U and V to be   U X V = {(u, v) : u is in U and vis in V} Prove that U X V is a vector space.

Linear Algebra: A Modern Introduction
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 49EQ
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If U and V are vector spaces, define the Cartesian product of U and V to be   U X V = {(u, v) : u is in U and vis in V} Prove that U X V is a vector space.

Expert Solution
Step 1

Given : U×V=u, v: u∈U, v∈V

To Prove: U×V is a vector space.

Step 2

U×V=u, v: u∈U, v∈V

(1) Let a=x, y, b=p, q , c=s, r

a+b+c=x+p, y+q+s, ra+b+c=x+p+s, y+q+ra+b+c=x, y+p+s, q+ra+b+c=a+b+c

U×V is associative under addition.

(2) Since 0∈U , 0∈V

⇒0, 0∈U×V

For any a∈U×V

a+0=0+a

 

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