Linear Algebra - VectorsPlease help me with this question! Thank you! Recall that Q is the set of rational numbers {" : m, n integers with n 0}. Consider thefollowing subset of real numbers:V := {a + bv2 : a, b e Q}.So V is the set of all real numbers that can be expressed in the form a + b/2 for rationalnumbers a and b.We define an addition operation on V by(a + bv2) + (c+ d/2) := (a + c) + (b + d)v2, for a + bv2, c+ dv2 € V.We define a scalar multiplication operation of the rationals Q by(a + bv2) := da,for a + bv2 E V and A E Q.(Note in particular that we are dealing with a scalar multiplication operation of Q, ratherthan the usual real numbers R.)(a) Show that V is closed under the addition and scalar multiplication defined above.(b) Is V a vector space over Q? If you think it is, you must prove all eight axioms. Ifnot, then show why one of the axioms doesn't hold.

The given set is,V=a+b2:a,b∈ℚThe addition and scalar multiplication on V is as…

Answered: We define an addition operation on V by…

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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Linear Algebra - Vectors

Please help me with this question! Thank you!

Recall that Q is the set of rational numbers {" : m, n integers with n 0}. Consider the
following subset of real numbers:
V := {a + bv2 : a, b e Q}.
So V is the set of all real numbers that can be expressed in the form a + b/2 for rational
numbers a and b.
We define an addition operation on V by
(a + bv2) + (c+ d/2) := (a + c) + (b + d)v2, for a + bv2, c+ dv2 € V.
We define a scalar multiplication operation of the rationals Q by
(a + bv2) := da,
for a + bv2 E V and A E Q.
(Note in particular that we are dealing with a scalar multiplication operation of Q, rather
than the usual real numbers R.)
(a) Show that V is closed under the addition and scalar multiplication defined above.
(b) Is V a vector space over Q? If you think it is, you must prove all eight axioms. If
not, then show why one of the axioms doesn't hold.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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