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Problem 5. Show that an element ū is a unit in R/PR, and find...
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u,...
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u, v e R". Show that ul if and only if ||ü + 해2 (c) Let W be a subspace of R" with an orthogonal basis {w1, ..., w,} and let {ö1, ..., ūg} 22 orthogonal basis for W- (i) Explain why{w1, ..., üp, T1, .., T,} is an (ii Explain why the set in (i) spans R". (iii Show that dim(W) + dim(W1) be...
(a) An element in a ring R is nilptent in there exists n e Z such...
(a) An element in a ring R is nilptent in there exists n e Z such that " = 0. The nilradical N of R is the set of its nilpotent elements. Find the nilradical of Z12. (b) Describe the ring Za[i]/(3+ i). (c) For which integers n does x2 + 2x + 1 divide x1 +52 + 2x2 + 3x + 15 in Z/nZ[:)?
Find a ring R with a proper ideal I and an element b such that b is not a unit in R but (b + I) is a unit in R/I.
Find a ring R with a proper ideal I and an element b such that b is not a unit in R but (b + I) is a unit in R/I.
(5) (10") Recall that for a unit vector ū in R, the matrix P = ūū...
(5) (10") Recall that for a unit vector ū in R, the matrix P = ūū represents the projection on ü. (a) Are there values a and b such that P is a SPD matrix? Explain. (b) Orthogonally diagonalize P. (c) Orthogonally diagonalize the reflection matrix L = 2P-I
37. Show that if D is an integral domain, then 0 is the only nilpotent element...
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
How can I find the inverse of the Linear Transformation From R^4 to R^4? x1 T...
How can I find the inverse of the Linear Transformation From R^4
to R^4?
x1 T = x1 22 -16 8 5 + x2 13 -3 9 4 x2 x3 x4 + x3 8 -2 7 3 + x4 3 -2 2 1
Problem 3.1. In R=2100 = {0, 1, ..., 98, 99), consider 17 and y = 15....
Problem 3.1. In R=2100 = {0, 1, ..., 98, 99), consider 17 and y = 15. (1) Find the inverse element of x in Z100 explicitly, if x E U(R). (2) Find a non-zero element z € 2100 such that yz - On and zy Or. Hint. Consider ged(17, 100) and god(15, 100). For (1), try to find m, n e Z such that 17m + 100n=1
2) (a) If P=r cost and r sint -2te' = 0, find dP/dt (b) If z=...
2) (a) If P=r cost and r sint -2te' = 0, find dP/dt (b) If z= cos(xy), show that x - y = 0 3) Find the maximum and minimum points of the functions in (a) and (b) (a) x2 - y2 + 2x - 4y +10 (b) x - y3 - 2xy +2 4) (a) Find the shortest distance from the origin to the the quadric surface 3x2 + y2 - 4xz= 4 (b) Find seadu dx Jx In(x+u)
Problem 5. Let f and g be R + R defined as f(x) = 2x +1...
Problem 5. Let f and g be R + R defined as f(x) = 2x +1 and g(x) = x3 – 2x + 1 Find go f and determine if it is bijective. If it is bijective find its inverse. (20 pts)
1. Let X~b(x; n, p) (a) For n 6, p .2, find () Prx> 3), (ii) Pr(x23), (ii) Pr(x (b) For n = 15, p= .8, find (i) Pr(X-2), (ii) Pr(X-12), (iii) Pr(X-8). (c) For n 10, find p so that Pr(X 2 8)6778....
1. Let X~b(x; n, p) (a) For n 6, p .2, find () Prx> 3), (ii) Pr(x23), (ii) Pr(x (b) For n = 15, p= .8, find (i) Pr(X-2), (ii) Pr(X-12), (iii) Pr(X-8). (c) For n 10, find p so that Pr(X 2 8)6778. く2). 2. Let X be a binomial random variable with μ-6 and σ2-2.4. Fin (a) Pr(X> 2) (b) Pr(2 < X < 8). (c) Pr(Xs 8).
1. Let X~b(x; n, p) (a) For n 6, p...
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u, v e R". Show that ul if and only if ||ü + 해2 (c) Let W be a subspace of R" with an orthogonal basis {w1, ..., w,} and let {ö1, ..., ūg} 22 orthogonal basis for W- (i) Explain why{w1, ..., üp, T1, .., T,} is an (ii Explain why the set in (i) spans R". (iii Show that dim(W) + dim(W1) be...
(a) An element in a ring R is nilptent in there exists n e Z such that " = 0. The nilradical N of R is the set of its nilpotent elements. Find the nilradical of Z12. (b) Describe the ring Za[i]/(3+ i). (c) For which integers n does x2 + 2x + 1 divide x1 +52 + 2x2 + 3x + 15 in Z/nZ[:)?
(5) (10") Recall that for a unit vector ū in R, the matrix P = ūū represents the projection on ü. (a) Are there values a and b such that P is a SPD matrix? Explain. (b) Orthogonally diagonalize P. (c) Orthogonally diagonalize the reflection matrix L = 2P-I
37. Show that if D is an integral domain, then 0 is the only nilpotent element in D. 38. Let a be a nilpotent element in a commutative ring R with unity. Show that (a) a = 0 or a is a zero divisor.. (b) ax is nilpotent for all x ER. (c) 1 + a is a unit in R. (d) If u is a unit in R, then u + a is also a unit in R.
How can I find the inverse of the Linear Transformation From R^4
to R^4?
x1 T = x1 22 -16 8 5 + x2 13 -3 9 4 x2 x3 x4 + x3 8 -2 7 3 + x4 3 -2 2 1
Problem 3.1. In R=2100 = {0, 1, ..., 98, 99), consider 17 and y = 15. (1) Find the inverse element of x in Z100 explicitly, if x E U(R). (2) Find a non-zero element z € 2100 such that yz - On and zy Or. Hint. Consider ged(17, 100) and god(15, 100). For (1), try to find m, n e Z such that 17m + 100n=1
2) (a) If P=r cost and r sint -2te' = 0, find dP/dt (b) If z= cos(xy), show that x - y = 0 3) Find the maximum and minimum points of the functions in (a) and (b) (a) x2 - y2 + 2x - 4y +10 (b) x - y3 - 2xy +2 4) (a) Find the shortest distance from the origin to the the quadric surface 3x2 + y2 - 4xz= 4 (b) Find seadu dx Jx In(x+u)
Problem 5. Let f and g be R + R defined as f(x) = 2x +1 and g(x) = x3 – 2x + 1 Find go f and determine if it is bijective. If it is bijective find its inverse. (20 pts)
1. Let X~b(x; n, p) (a) For n 6, p .2, find () Prx> 3), (ii) Pr(x23), (ii) Pr(x (b) For n = 15, p= .8, find (i) Pr(X-2), (ii) Pr(X-12), (iii) Pr(X-8). (c) For n 10, find p so that Pr(X 2 8)6778. く2). 2. Let X be a binomial random variable with μ-6 and σ2-2.4. Fin (a) Pr(X> 2) (b) Pr(2 < X < 8). (c) Pr(Xs 8).
1. Let X~b(x; n, p) (a) For n 6, p...
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