Let M2x2(F) be the set of all 2 x 2 matrices with entries in a field F with addition and scalar multiplication defined as follows: b b+y • (ad) + ( # ²) = (a + x à tg). for a, b, c, d, x, y, w, z in F • b a(d) = (ac ad) aa ab for a, a, b, c, d in F.

Let M2x2(F) be the set of all 2 x 2 matrices with entries in a field F with addition and scalar multiplication defined as follows: b b+y • (ad) + ( # ²) = (a + x à tg). for a, b, c, d, x, y, w, z in F • b a(d) = (ac ad) aa ab for a, a, b, c, d in F.

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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Question

Prove that this is a vector space over F by checking that all of the 4 axioms of a vector space holds . Indicate for each of the 4 axioms of vector spaces which of the 5 axioms of the field F you are using in your calculations. Yo umay also use the following fact (which is a consequence of the axioms offield) without proving it:

(⋆) for any element α ∈ F , 0 · α = 0

Let M2x2(F) be the set of all 2 x 2 matrices with entries in a field F with
addition and scalar multiplication defined as follows:
b
x
b
(a d) + (+ ²) = (a + d + ²)
(₁
1
for a, b, c, d, x, y, w, z in F
• a (a
x d) = (a ad). for a, a, b, c, d in F.
ab

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