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Statement Let V be any vector space and let x, y, z be in V. Prove that if x + z = y +z, then x = y. Proof piem Suppose V is any vector space and x, y, z are any vectors in V. Suppose x + z = y + z. Show that x = y Since z is in V, there exists a vector -z in V such that z + (-z) = 0 Thus, x + z = y + z (x+z) + (-z) = (y+z) + (-z) x + (z + (−z)) = y + (z+ (−z)) because x + 0 = y +0 because and thus x = y Therefore, x + z = y + z implies x = y, and V has the cancellation property. Allowed answers because addition is commutative implies and thus by properties of the zero vector. V is a vector space, V is closed under addition, V is closed under scalar multiplication, addition is commutative in V, addition is associative in V, the zero vector in V is an additive identity, additive inverses exist in V, 1 is a multiplicative identity, scalar multiplication is associative, scalar multiplication distributes over vector addition, scalar multiplication distributes over scalar addition

Statement Let V be any vector space and let x, y, z be in V. Prove that if x + z = y +z, then x = y. Proof piem Suppose V is any vector space and x, y, z are any vectors in V. Suppose x + z = y + z. Show that x = y Since z is in V, there exists a vector -z in V such that z + (-z) = 0 Thus, x + z = y + z (x+z) + (-z) = (y+z) + (-z) x + (z + (−z)) = y + (z+ (−z)) because x + 0 = y +0 because and thus x = y Therefore, x + z = y + z implies x = y, and V has the cancellation property. Allowed answers because addition is commutative implies and thus by properties of the zero vector. V is a vector space, V is closed under addition, V is closed under scalar multiplication, addition is commutative in V, addition is associative in V, the zero vector in V is an additive identity, additive inverses exist in V, 1 is a multiplicative identity, scalar multiplication is associative, scalar multiplication distributes over vector addition, scalar multiplication distributes over scalar addition

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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Statement Let V be any vector space and let x, y, z be in V. Prove that if x + z = y + z, then x = y. Proof Text Problem Suppose V is any vector space and x, y, z are any vectors in V. Suppose x + z = y + z. Show that x = y Since z is in V, there exists a vector -z in V such that z + (-z) = 0 Thus, x + z = y + z (x+z) + (-z) = (y+z) + (-z) x + (z + (−z)) = y + (z+ (-z)) because x+0=y+0 and thus x = y Therefore, x + z = y + z implies x = y, and V has the cancellation property. Allowed answers because addition is commutative implies because and thus by properties of the zero vector. V is a vector space, V is closed under addition, V is closed under scalar multiplication, addition is commutative in V, addition is associative in V, the zero vector in V is an additive identity, additive inverses exist in V, 1 is a multiplicative identity, scalar multiplication is associative, scalar multiplication distributes over vector addition, scalar multiplication distributes over scalar addition
Transcribed Image Text:Statement Let V be any vector space and let x, y, z be in V. Prove that if x + z = y + z, then x = y. Proof Text Problem Suppose V is any vector space and x, y, z are any vectors in V. Suppose x + z = y + z. Show that x = y Since z is in V, there exists a vector -z in V such that z + (-z) = 0 Thus, x + z = y + z (x+z) + (-z) = (y+z) + (-z) x + (z + (−z)) = y + (z+ (-z)) because x+0=y+0 and thus x = y Therefore, x + z = y + z implies x = y, and V has the cancellation property. Allowed answers because addition is commutative implies because and thus by properties of the zero vector. V is a vector space, V is closed under addition, V is closed under scalar multiplication, addition is commutative in V, addition is associative in V, the zero vector in V is an additive identity, additive inverses exist in V, 1 is a multiplicative identity, scalar multiplication is associative, scalar multiplication distributes over vector addition, scalar multiplication distributes over scalar addition
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