oach of this section is certainly not the only way to construct C. We could also view C as a special set of 2 x 2 matrices. (We will not burden ourselves by giving a formal definition of a matrix.) (i) Check that is really matrix-vector multiplication: (x, y) (u, v) = (xu-yv, xv + yu) (ii) Now put = R2X² :=< [xu- yv] -Y [+][4]0-30) v xv + yu = y = U V ~](). = { [~ 7] | ₁₁ VER} x, and check that C (as defined in Definition 1.4.5) and R2x2 are isomorphic as fields, where
oach of this section is certainly not the only way to construct C. We could also view C as a special set of 2 x 2 matrices. (We will not burden ourselves by giving a formal definition of a matrix.) (i) Check that is really matrix-vector multiplication: (x, y) (u, v) = (xu-yv, xv + yu) (ii) Now put = R2X² :=< [xu- yv] -Y [+][4]0-30) v xv + yu = y = U V ~](). = { [~ 7] | ₁₁ VER} x, and check that C (as defined in Definition 1.4.5) and R2x2 are isomorphic as fields, where
Chapter7: Systems Of Equations And Inequalities
Section7.7: Solving Systems With Inverses
Problem 1SE: In a previous section, we showed that matrix multiplication is not commutative, that is, ABBA in...
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Question
Use Definition 1.4.5 to solve 1.4.13
![1.4.13 Problem (P). (Presumes knowledge of linear and abstract algebra.) The approach
of this section is certainly not the only way to construct C. We could also view C as a
special set of 2 x 2 matrices. (We will not burden ourselves by giving a formal definition
of a matrix.)
(i) Check that is really matrix-vector multiplication:
(x, y) • (u, v) = (xu-yv, xv + yu) =
(ii) Now put
[+][10-30
1.5. Functions
xu - yv
xv + yu
=
=
R²X² := {[17] |
{[t −7] | T, VER}
x, y
and check that C (as defined in Definition 1.4.5) and R2ײ are isomorphic as fields, where
27
addition and multiplication in Care and Ⓒ, while addition and multiplication in R2x2
are the usual addition and multiplication for 2 × 2 matrices.
(iii) To what subset of R2² is R (as defined in Definition 1.4.5) isomorphic as a field?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2Fad2e6d99-4796-45d1-9558-7eddb62efa6e%2Ftrhfzcd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.4.13 Problem (P). (Presumes knowledge of linear and abstract algebra.) The approach
of this section is certainly not the only way to construct C. We could also view C as a
special set of 2 x 2 matrices. (We will not burden ourselves by giving a formal definition
of a matrix.)
(i) Check that is really matrix-vector multiplication:
(x, y) • (u, v) = (xu-yv, xv + yu) =
(ii) Now put
[+][10-30
1.5. Functions
xu - yv
xv + yu
=
=
R²X² := {[17] |
{[t −7] | T, VER}
x, y
and check that C (as defined in Definition 1.4.5) and R2ײ are isomorphic as fields, where
27
addition and multiplication in Care and Ⓒ, while addition and multiplication in R2x2
are the usual addition and multiplication for 2 × 2 matrices.
(iii) To what subset of R2² is R (as defined in Definition 1.4.5) isomorphic as a field?
Transcribed Image Text:1.4.5 Definition. (i) A COMPLEX NUMBER is an ordered pair (x,y), where x, y ≤ R.
1.4. Rigorous constructions of the complex numbers
We denote the set of all complex number numbers by
C = {(x,y) | x, y ≤ Ⓡ}.
24
(ii) A REAL NUMBER is an ordered pair (x,0), where x ER. We denote the set of all
real numbers by
R={(1,0) | TER}.
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