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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 Suppose for full generality that 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+(-2) = 0 because implies Thus, x +z = y + z (x + 2) + (-2) = (y + 2) + (-2) and thus x+(2+(-2)) = y + (z + (-2)) because X+0 = y + 0 because and thus X = y by properties of the zero vector. Therefore, x +z = y + z implies x = y, and V has the cancellation property

Respuesta :

By the zero vector property x=y.

Suppose for full generality that V is any vector space and x,y,z is any vector 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, because x+y=y+z

Thus x+z=y+z

= (x+z)+(-z) = (y+z)+(-z) and thus

= x+(z+(-z)) = y + (z+(-z)) because addition is associative in vector

= x+0 = y+0 because additive inverse exists in a vector

= and thus x=y by the properties of zero vector

Also according to the question, it is now proved that x+z=y+z implies x=y and v has the cancellation property.


To learn more about vectors and their properties,

https://brainly.com/question/13322477

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