Which of the following statements are true, and which are false? Give reasons for your answers.
i) For any ring R and .
ii) Every ring has at least two elements.
iii) If R is a ring with identity and I is an ideal of R, then the identity of R/I is the same as the identity of R.
iv) If is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +).
v) If R is a ring, then any ring homomorphism from into R is surjective.
i) False – Not every equation over a ring R has a solution. For example, in the ring of integers ℤ, the equation 2x = 1 has no solution in ℤ. ii) False – ___ _________ ___ _____ _______ _____.
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________ _________ ____ _______ ____ ________ ________ _________ _____.
____ _____ ___ _____ ___ ______ _______ ___.
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की अभिसरण कोटि निर्धारित कीजिए।
Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then .
ii) If x and y are elements of a non-abelian group (G, ) such that , then
or
, where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If , where A is a group, then
.
v) If H and K are normal subgroups of a group G, then .
Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix using the power method. Take (1, 0, 0)T as the initial approximation and perform 4 iterations.
Obtain the interpolating polynomial in simplest form which fits the following data:
| x | -1 | 0 | 1 | 2 |
| f(x) | 3 | -4 | 5 | -6 |
निम्नलिखित तालिका में लुप्त मान ज्ञात कीजिए :
Find the approximate root of the equation using Newton-Raphson method. Perform only 3 iterations with
.
If , find the values of C1 and C2.