16.1.14. Let R = M2×2 (R) be the ring of two-by-two matrices with real entries.[0 b[ab]Let S = {| a, b = R} and T={0 a00[o of] b € R} .(a) Are T and S subrings of R?(b) Is T an ideal of S? Is T an ideal of R? Is S an ideal of R?

Step 1: Part (a): Are T and S subrings of R?For S: Closure under addition:If A,B∈S, then A+B∈S since…

Answered: 16.1.14. Let R = M2×2 (R) be the ring…

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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Which of the following six sets are subrings of M(R)? Which ones have an identity? 0 0 (a) All matrices of the form (8 5) with reQ. (b) All matrices of the form (c) All matrices of the form (d) All matrices of the form (a b) 0 Ca a a with a, b, cЄZ. b with a, b, c = R. 0 with a Є R. 0 with a ЄR. (e) All matrices of the form a (69) (f) All matrices of the form 0 (a) with a ЄR. 0
* Consider the basis .3 B = {1 + x + x², x + x², x²}. for P2. Let p(x) 1 + 3x + 2x2. Then [p(x)]B 1 2 None O 2
A transformation is defined by the 2 × 2 matrixA =−a b − aa + b a where a and b are scalars. If n is an odd integer, prove thatAn = bn−1A
Let B={1, x, x², x³ be a basis for P3, and let P={P1, P2, P3, P4} be the set of polynomials given below: P₁(x) = 2x³+3x²+2x−1 P2(x) = −2x³–3x²+2x P3(x) = −2x²-x+1 P4(x) = −2x³+x+2 Find the coordinates of each of these polynomials with respect to the basis B, and use the coordinate vectors to determine whether P is linearly independent, and whether it spans P3. 0 0 0 0 [P1(x)]B= 0 [P2(x)]B= 0 [P3(x)]B = [P4(x)]B = The set P is linearly independent The set P spans P3
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T= Consider the subrings а C { [° d]|a,c,d € Q} C, 0 and U = { [88]|₁ € Q} of the matrix ring M₂(Q). Show that U is an ideal in the subring T but not an ideal in M₂(Q).
3: Which of the following sets are closed under scalar multiplication? (1) The set of all vectors in R of the form (a, b, 9ab). (i1) The set of all polynomials a- bx+ cx in P, such that ac =D0. (iii) The set of all matrices in Mn such that A = A (A) (1) and (ii) only (B) (1) and (iii) only (C) (ii) and (iii) only (D) (1) only (E) (ii) only (F) none of them (G) (iii) only (H) all of them
Show that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.
14. Consider the basis B = {1 +x + x², x + x², x²}. for P2. Let p(x) = 2 + 3x + 2x². Then [p(x)]B 2 2 O None 1 2
Drag each of the six matrices to the column above which the polynomial corresponds to its characteristic polynomial. 2 1 1 0 0 -4 5 0 1 4 2 2 -5 -6 -5 7 5 8 -x³ +5x² - 2x - 8 9 6 -13 -10 8 6 9 4 7 -7 16 -24 -3 4 0 (AD CAD CD 5 4 5 4 -3 8 9 8 -10 -4 -8 4 -8 13 0 0 -3 -r³+3x²+ 13x - 15 0 -³x² + 10x - 8 C Reset
Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||p|, |la|l, and d(p, q) for the polynomials in P₂. p(x) = 1 -x + 4x², g(x) = x - x² (a) (p, q) (b) ||p|| (c) |||| (d) d(p, q)
LetT: P₂ → P3 be the transformation that maps a polynomial p(t) into the polynomial (5t + 2)p(t)) Find the matrix for T relative to the following bases B and C. C={C₁,C₂,C3,C4} = {1,t,t²,t³} B=(b₁,b₂,b3} = {1,t,t²}, 500 250 025 002 a. b. C. d. 5200 05 20 0052 200 520 052 005 2500 0250 0025

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16.1.14. Let R = M2×2 (R) be the ring of two-by-two matrices with real entries. [0 b [a b] Let S = { | a, b = R} and T = { 0 a 00 [o of] b € R} . (a) Are T and S subrings of R? (b) Is T an ideal of S? Is T an ideal of R? Is S an ideal of R?