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 ___ ______ ______ _______ ________ _______ ____ ___ _____ ______ _________ ______.
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Write the error in linear interpolation. Hence, show that
where
गर्शगोरिन परिबंधों का प्रयोग करके आव्यूह$
के आइगेनमान आकलित कीजिए।
Find all the units of .
State whether the following statements are true or false. Give a Short proof or a counter-example in support of your answer.
a) The equation has not root in the interval [3, 5].
b) , where E is the shift operator and
is the forward difference operator.
c) Every system of linear equations can be solved using the LU decomposition method.
d) For the data (2, 4), (1, 5), (3, 6) the Newton's divided difference f[x0, x1, x2] is .
e) The Newton-Raphson method cannot be used to find a cube root of a positive real number.
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.
Find the interpolating polynomial by Newton’s divided difference formula for the following data:
| x | 0 | 1 | 2 | 4 |
| y | 1 | 1 | 2 | 5 |
विधि$
जहाँ N एक धन अचर है, N1/3 की ओर अभिसरित होती है। विधि की अभिसरण दर ज्ञात कीजिए।