Question

Let R = M(2, Q), the ring of 2 × 2 matrices with rational entries. a) Check that the subset of R Ru = {(? a b ? | a, b, c ? Q) ? 0 c ?} consisting of upper triangular matrices is in fact a subring of R. b) Let Rl = {(? d 0 ? | d, e, f ? Q) ? e f ?} be the subring of lower triangular matrices. (You don't have to check that this is a subring.) Show that Ru is isomorphic to Rl. You should define a function ? : Ru ? Rl and prove that ? is a ring homomorphism and a bijection.

          Let R = M(2, Q), the ring of 2 × 2 matrices with rational entries.
a) Check that the subset of R
Ru = {(? a b ? | a, b, c ? Q)
? 0 c ?}
consisting of upper triangular matrices is in fact a subring of R.
b) Let
Rl = {(? d 0 ? | d, e, f ? Q)
? e f ?}
be the subring of lower triangular matrices. (You don't have to check that this is a subring.)
Show that Ru is isomorphic to Rl. You should define a function ? : Ru ? Rl and prove that ? is a ring homomorphism and a bijection.

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Let R = M(2, Q), the ring of 2 × 2 matrices with rational entries.
a) Check that the subset of R
Ru = (? a b ? | a, b, c ? Q)
? 0 c ?
consisting of upper triangular matrices is in fact a subring of R.
b) Let
Rl = (? d 0 ? | d, e, f ? Q)
? e f ?
be the subring of lower triangular matrices. (You don't have to check that this is a subring.)
Show that Ru is isomorphic to Rl. You should define a function ? : Ru ? Rl and prove that ? is a ring homomorphism and a bijection.

Added by Erin P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

05/09/2023

Step 1

First, we need to show that the subset $Ru$ of upper triangular matrices is a subring of $R$. To do this, we need to show that it is closed under addition and multiplication, and that it contains the identity matrix and additive inverses. a) Let $A = Show more…

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Let R = M(2, Q), the ring of 2 x 2 matrices with rational entries. a) Check that the subset of R Ru = {(a b | 0 c) | a, b, c ∈ Q} consisting of upper triangular matrices is in fact a subring of R. b) Let Rl = {(d 0 | e f) | d, e, f ∈ Q} be the subring of lower triangular matrices. Show that Ru is isomorphic to Rl. You should define a function φ : Ru → Rl and prove that φ is a ring homomorphism and a bijection.

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