Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is .
ii) If R is an integral domain and I is an ideal of R, then .
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R, then S has zero divisors.
v) If R is a ring and is of degree
, then f(x) has exactly n roots in R.
i) False. Just because (k) is a field does not automatically make an equation a field. An equation is a relation, not a set with field operations. Fields are defined as sets with addition and multiplication satisfying certain axioms, so the statement is incorrect. ii) False. If (R) is an integral domain and (I) is an ___ ___ ________ ____ _____ _______ ____.
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The Gauss-Seidel method is used to solve the system of equations
Determine the rate of convergence of the method.
Find all the units of .
गर्शगोरिन परिबंधों का प्रयोग करके आव्यूह$
के आइगेनमान आकलित कीजिए।
For an ideal I of a commutative ring R, define. Show that
i) is an ideal of R.
ii) .
iii) in some cases.
) Evaluate by using trapezoidal rule with
and
. Use Romber's method to find the best value of
.
) Estimate the eigenvalues of the matrix
using the Gerschgorin bounds.
निम्नलिखित तालिका में लुप्त मान ज्ञात कीजिए :
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.