1.a) Let L be the subset of M (IR) consisting of matrices of the form [0is a subring of M(R).[]. Prove that Lb) Let U be the subset of M(R) consisting of matrices of the form [is a subring of M(R).[6]. Prove that Uc) Is UUL a subring of M(R)? Prove it or provide a counterexample.d) Find Un L. Is Un La subring of M(R)? Justify your answer.e) Prove the following: If S and T are subrings of a ring R, then SnT is a subring of R.

//The answer above is the detailed solution of the given problem //Feel free to ask doubts if…

Answered: 1. a) Let L be the subset of M (IR)…

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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Let B = {v1, V2, V3} be a basis for R3. For any u, v E R³ define || (u, v) = [u]E[v]B. • (a) Is (,) an inner product? (b) Can you find a matrix A (constructed using v1, V2, and v3) such that (u, v) = u" Av?
3) Let Z, be ring integers mode 5 and f(x) € Zs(x) such that f(x) = x² + 2x + 1 then a) f(x) has only one root. b) f(x) has only two roots. c) f(x) irreducible polynomials.. d) No one of above. 4) Let R be a ring and F-M₂(R) be ring of 2 x 2 matrices over R under standard addition and multiplication of matrices then: a) F is a commutative ring but not field. b) F is a division ring and not field. (c) If F an integral domain and F is not field. d) No one of above.
Show that the set of all matrices of order 2×2 of the form [10], where a, b, c are integers is a subring of the ring M of all matrices of order 2 × 2 with integral elements.
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3 Define the set S of matrices by S = {A = (aij) = M₂ (R) : a11 = a22, a12 = −a21}. It turns out that S is a ring, with the operations of matrix addition and multiplication. (a) Write down two examples of elements of S, and compute their sum and product (b) Prove the additive and multiplicative closure laws for S.
4. Let V = {ao+aj x+a2 x² | ao, a1, a2 E R} be a vector space of polynomial up to degree three, and p(x) = pPo+P1 x+p2 x², q(x) = qo+q1 x+q2 x² E V then show that = po.4o + pP1-q1 + P2-92 defines an inner product on V.
Suppose that β and γ are ordered bases for an n-dimensional real [complex] inner product space V. Prove that if Q is an orthogonal [unitary] n ×n matrix that changes γ-coordinates into β-coordinates, then β is orthonormal if and only if γ is orthonormal.
4. Define the following complex vector space = a € = (²x² = a +d=0} with the usual addition and scalar multiplication. (a) Find an appropriate value of r such that V~ Cr. (b) Give an explicit isomorphism T: V→ Cr. Make sure to prove that T is an isomorphism!
Let K = Z5, the field of integers modulo 5. (You can read about fields in Chapter 1.8 of the textbook). Consider the vector space P2 of polynomials of degree at most 2 with coefficients in K. Are the polynomials x + x + 3, 4x² + 2x + 4, and 4x + 1 linearly independent over Z5? linearly independent v If they are linearly dependent, enter a non-trivial solution to the equation below. If they are linearly independent, enter any solution to the equation below. (х2 + х + 3)+ о (4x2 + 2x + 4)+ 0 (4х + 1) — 0.
Please give as much detail as possible
Which of the following statements is (are) true if B is a basis for the vector space V (a) In some situations, a plane in R6 can be isomorphic to R4 (b) If x is in V and B contains n vectors, the the B- coordinate vector of x is in R (c) The vector spaces P5 and RS are isomorphic. (d) If PB is the change of coordinates matrix, then [x]B = PB x for x in V (e) A coordinate mapping is a bijective mapping O All of the above Noneoftheabove Options a, b, c and e Options a, b and e Options b, c and e Options a, b, d and e Options b, c, d and e Options a, and b
Let T: R² R² be defined by T and C Given Pc 3 -{]-[H]} -1223) , use the Fundamental Theorem of Matrix Representations to find [T](PB(u)). [T](PB (u)) 7 [4₁]) B-{B] R]} = Let u = = Ex: 5 [-3x1 + 2x₂ / -4

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