Answered: If U and V are vector spaces, define…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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 : $U \times V = \{(u, v) : u \in U, v \in V\}$

To Prove: $U \times V$ is a vector space.

Step 2

$U \times V = \{(u, v) : u \in U, v \in V\}$

(1) Let $a = (x, y), b = (p, q), c = (s, r)$

$(a + b) + c = (x + p, y + q) + (s, r) (a + b) + c = (x + p + s, y + q + r) (a + b) + c = (x, y) + (p + s, q + r) (a + b) + c = a + (b + c)$

$U \times V$ is associative under addition.

(2) Since $0 \in U, 0 \in V$

$\Rightarrow (0, 0) \in U \times V$

For any $a \in U \times V$

$a + 0 = 0 + a$

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