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12.4 (12.4) Find the transformation properties of E, B, p, and j under a boost in...
8. Question from 12.4: The Cross Product Find the vector, not with determinants, but by using...
8. Question from 12.4: The Cross Product Find the vector, not with determinants, but by using properties of cross products. (i x j) x k
Exercise 3. (12p) (Lorentz boosts) The Maxwell equations (7) are invariant under Lorentz transformations. This implies...
Exercise 3. (12p) (Lorentz boosts) The Maxwell equations (7) are invariant under Lorentz transformations. This implies that given a solution of the Maxwell equa- tions, we obtain another solution by performing a Lorentz transformation to the solution. A particular Lorentz transformation is a Lorentz boost with velocity v in - direction and acts on the electric and magnetic field strength as given in appendix B. (1) Tong) Now consider the electric and magnetic field due to a line along the...
1. Derive the Lorentz transformation for the "3-vectors" É and B from the normal orentz transtormation for the ran (Helps to get it set up like a normal matrix multiplication.) Give the t...
1. Derive the Lorentz transformation for the "3-vectors" É and B from the normal orentz transtormation for the ran (Helps to get it set up like a normal matrix multiplication.) Give the transforma tion for the components of the 3-vectors parallel to the boost and perpendicular to the boost. Hint: Do the calculation of F' by multiplying the matrices. Then look at compo- nents of the fields parallel and perpendicular to the boost direction. k 2 held tensor r.μν-αμα avprop-αμα...
6 image of x under 4. Problem 2: With the transformation T(x)- AX, and the vector b T, find if it exists. (13 points...
6 image of x under 4. Problem 2: With the transformation T(x)- AX, and the vector b T, find if it exists. (13 points). Is the transformation one-to-one or onto, neither or both? Justify your answer. (7 points).
6 image of x under 4. Problem 2: With the transformation T(x)- AX, and the vector b T, find if it exists. (13 points). Is the transformation one-to-one or onto, neither or both? Justify your answer. (7 points).
P.2.16 Let V= span {AB-BA : A, B E Mn. (a) Show that the function tr...
P.2.16 Let V= span {AB-BA : A, B E Mn. (a) Show that the function tr : M,,-> C is a linear transformation. (b) Use the dimension theorem to prove that dim ker tr = n2-1. (c) Prove that dim V = n2-1. (d) Let Eij=eie), every entry of which is zero except for a 1 in the (i, j) position. Show that k,-OikEil for l i, j, k, n. (e) Find a basis for V. Hint: Work out the...
10. Consider the basis for P, »{1,x,x+,x"}. Let T be a transformation T:P, , where T(x*)=...
10. Consider the basis for P, »{1,x,x+,x"}. Let T be a transformation T:P, , where T(x*)= *t* dt. Find a standard matrix for this transformation. (Hint: You may need to review calculus and think about how P. polynomials can be represented as R"+1 vectors. n
Let T:P R^2 be defined by T(p(x)) = (p(1),p(-1)). (a) Find T(p(x)) where p(x) = 2 + 5x. (b) Show that T is a linear transformation. (C) Find the kernel of T. Explain why T is one-to-one. (d) Find the range of T. Explain why I' is onto. (e) Find T-1(3,7)
Let T:P R^2 be defined by T(p(x)) = (p(1),p(-1)). (a) Find T(p(x)) where p(x) = 2 + 5x. (b) Show that T is a linear transformation. (C) Find the kernel of T. Explain why T is one-to-one. (d) Find the range of T. Explain why I' is onto. (e) Find T-1(3,7)
Consider the transformation T[x y] = [x + y y^2] a. Is T a linear transformation?...
Consider the transformation T[x y] = [x + y y^2]
a. Is T a linear transformation?
b. Is the range of T closed under addition?
c. "" scalar multiplication?
10. Consider the transformation T1yHyy (a) Is Ta linear transformation? (b) Is the range of T closed under addition? (e) Is the range on T closed under scalar multiplication?
Consider the the transformation T: P^2 -> T^2 defined by T(a+bx+cx^2) = (a+b, b-c). Find the...
Consider the the transformation T: P^2 -> T^2 defined by T(a+bx+cx^2) = (a+b, b-c). Find the kernel of T. Give 2 examples of vectors in the kernel.
Define the linear transformation T:?3??4 by T(x )=Ax . Find a vector x whose image under...
Define the linear transformation T:?3??4 by T(x )=Ax . Find a
vector x whose image under T is b
(1 pt) Let 4 5 2 -2 5 -3 2 and b-10 -7 2 1 -4 Define the linear transformation T : R3 ? R4 by T(x-Ax Find a vector x whose image under T is b. x= Is the vectorx unique? choose
8. Question from 12.4: The Cross Product Find the vector, not with determinants, but by using properties of cross products. (i x j) x k
Exercise 3. (12p) (Lorentz boosts) The Maxwell equations (7) are invariant under Lorentz transformations. This implies that given a solution of the Maxwell equa- tions, we obtain another solution by performing a Lorentz transformation to the solution. A particular Lorentz transformation is a Lorentz boost with velocity v in - direction and acts on the electric and magnetic field strength as given in appendix B. (1) Tong) Now consider the electric and magnetic field due to a line along the...
1. Derive the Lorentz transformation for the "3-vectors" É and B from the normal orentz transtormation for the ran (Helps to get it set up like a normal matrix multiplication.) Give the transforma tion for the components of the 3-vectors parallel to the boost and perpendicular to the boost. Hint: Do the calculation of F' by multiplying the matrices. Then look at compo- nents of the fields parallel and perpendicular to the boost direction. k 2 held tensor r.μν-αμα avprop-αμα...
6 image of x under 4. Problem 2: With the transformation T(x)- AX, and the vector b T, find if it exists. (13 points). Is the transformation one-to-one or onto, neither or both? Justify your answer. (7 points).
6 image of x under 4. Problem 2: With the transformation T(x)- AX, and the vector b T, find if it exists. (13 points). Is the transformation one-to-one or onto, neither or both? Justify your answer. (7 points).
P.2.16 Let V= span {AB-BA : A, B E Mn. (a) Show that the function tr : M,,-> C is a linear transformation. (b) Use the dimension theorem to prove that dim ker tr = n2-1. (c) Prove that dim V = n2-1. (d) Let Eij=eie), every entry of which is zero except for a 1 in the (i, j) position. Show that k,-OikEil for l i, j, k, n. (e) Find a basis for V. Hint: Work out the...
10. Consider the basis for P, »{1,x,x+,x"}. Let T be a transformation T:P, , where T(x*)= *t* dt. Find a standard matrix for this transformation. (Hint: You may need to review calculus and think about how P. polynomials can be represented as R"+1 vectors. n
Consider the transformation T[x y] = [x + y y^2]
a. Is T a linear transformation?
b. Is the range of T closed under addition?
c. "" scalar multiplication?
10. Consider the transformation T1yHyy (a) Is Ta linear transformation? (b) Is the range of T closed under addition? (e) Is the range on T closed under scalar multiplication?
Define the linear transformation T:?3??4 by T(x )=Ax . Find a
vector x whose image under T is b
(1 pt) Let 4 5 2 -2 5 -3 2 and b-10 -7 2 1 -4 Define the linear transformation T : R3 ? R4 by T(x-Ax Find a vector x whose image under T is b. x= Is the vectorx unique? choose
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