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7. For each linear operator T find a formula for T*. (a) Find T*(91: y2) for...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
(1 point) Suppose y2 y2 3 3 2 2 y(t) = cie + cze [1] 1...
(1 point) Suppose y2 y2 3 3 2 2 y(t) = cie + cze [1] 1 y1 y1 [-1] 7(1) = [21] -3 -2 2 3 -3 -2 -1 1 2 3 -1 -2 -3 -3 (a) Find ci and C2 А B C1 = 1.3591 y2 y2 3 3 C2 = 0.1839 2 2 1 1 y1 y1 -2 -1 1 2 3 -3 -2 -1 1 2 3 (b) Sketch the phase plane trajectory that satisfies the given...
(1 point) Suppose y2 y2 3 3 2 2 y(t) = cie + cze [1] 1...
(1 point) Suppose y2 y2 3 3 2 2 y(t) = cie + cze [1] 1 y1 y1 [-1] 7(1) = [21] -3 -2 2 3 -3 -2 -1 1 2 3 -1 -2 -3 -3 (a) Find ci and C2 А B C1 = 1.3591 y2 y2 3 3 C2 = 0.1839 2 2 1 1 y1 y1 -2 -1 1 2 3 -3 -2 -1 1 2 3 (b) Sketch the phase plane trajectory that satisfies the given...
2. Let T be the linear operator on C2 defined by Tc? = (1 + ?,...
2. Let T be the linear operator on C2 defined by Tc? = (1 + ?, 2), Te,-(i, i). Using the standard inner product, find the matrix of T* in the standard ordered basis. Does T commute with T*?
n Exercises 15–16, find the eigenvalues and a basis for each eigenspace of the linear operator...
n Exercises 15–16, find the eigenvalues and a basis for each
eigenspace of the linear operator defined by the stated formula.
[Suggestion: Work with the standard matrix for the operator.]
16. T(x,y,z)=(2x−y−z,x−z,−x+y+2z)
In Exercises 15-16, find the eigenvalues and a basis for each eigenspace of the linear operator defined by the stated formula Suggestion: Work with the standard matrix for the operator) 16. T(x, y, z) = (2x - y - 3. - 3. -* + y + 22)
8. Find the standard matrix representation for each linear operator L: R2 + R2 described below:...
8. Find the standard matrix representation for each linear operator L: R2 + R2 described below: (a) L rotates each vector 7 by 45° in the clockwise direction. (b) L reflects each vector 7 about the 21 axis and then rotates it 90° in the counterclockwise direction. (c) L doubles the length of t and then rotates it 30° in the counterclockwise direction. (d) L reflects each vector 7 about the line x2 = 21 and projects it onto the...
parts a, b, c and d . (a) T(31,72) = (2x - 20, -2.61 +5r) on...
parts a, b, c and d
.
(a) T(31,72) = (2x - 20, -2.61 +5r) on V =R? (b) T(31,12,13) = (-11 + 12,562, 46, -212 +503) on V =R3. (c) T(21, 12) = (2x1 + ix2, 21 +222) on V = C. d] on V = M2x2(R) (with the Frobenius Inner Product). с (d) T ] 3. For each linear operator T in Exercise 1 find an ON basis 3 for V consisting of eigenvectors of T (if possible).
3)For each linear operator T, find a basis for each generalized eigenspace of T consisting of a union of disjoint cycles of generalized eigenvectors. Then find a Jordan conical form J of T. T is the l...
3)For each linear operator T, find a basis for each generalized eigenspace of T consisting of a union of disjoint cycles of generalized eigenvectors. Then find a Jordan conical form J of T. T is the linear operator on M2x2 defined by T(A) = 1 1 0 1 * A for all A in M2x2(R)
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 ,...
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 , and y3(z) = eェamong its solutions. (b) Now find a different linear homogeneous differential equation of an order lower than the one in (a) that has the same y1,U2,U3 among its solutions. (c) Find a constant coefficient linear homogeneous differential equation of lowest order that...
In Exercise, find the eigenvalues of each linear operator and determine a basis for each eigenspace....
In Exercise, find the eigenvalues of each linear operator and determine a basis for each eigenspace. T -6x1 - 5x2 + 5x37 - 12 L-10x1 - 10x2 + 9x3]
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
(1 point) Suppose y2 y2 3 3 2 2 y(t) = cie + cze [1] 1 y1 y1 [-1] 7(1) = [21] -3 -2 2 3 -3 -2 -1 1 2 3 -1 -2 -3 -3 (a) Find ci and C2 А B C1 = 1.3591 y2 y2 3 3 C2 = 0.1839 2 2 1 1 y1 y1 -2 -1 1 2 3 -3 -2 -1 1 2 3 (b) Sketch the phase plane trajectory that satisfies the given...
(1 point) Suppose y2 y2 3 3 2 2 y(t) = cie + cze [1] 1 y1 y1 [-1] 7(1) = [21] -3 -2 2 3 -3 -2 -1 1 2 3 -1 -2 -3 -3 (a) Find ci and C2 А B C1 = 1.3591 y2 y2 3 3 C2 = 0.1839 2 2 1 1 y1 y1 -2 -1 1 2 3 -3 -2 -1 1 2 3 (b) Sketch the phase plane trajectory that satisfies the given...
2. Let T be the linear operator on C2 defined by Tc? = (1 + ?, 2), Te,-(i, i). Using the standard inner product, find the matrix of T* in the standard ordered basis. Does T commute with T*?
n Exercises 15–16, find the eigenvalues and a basis for each
eigenspace of the linear operator defined by the stated formula.
[Suggestion: Work with the standard matrix for the operator.]
16. T(x,y,z)=(2x−y−z,x−z,−x+y+2z)
In Exercises 15-16, find the eigenvalues and a basis for each eigenspace of the linear operator defined by the stated formula Suggestion: Work with the standard matrix for the operator) 16. T(x, y, z) = (2x - y - 3. - 3. -* + y + 22)
8. Find the standard matrix representation for each linear operator L: R2 + R2 described below: (a) L rotates each vector 7 by 45° in the clockwise direction. (b) L reflects each vector 7 about the 21 axis and then rotates it 90° in the counterclockwise direction. (c) L doubles the length of t and then rotates it 30° in the counterclockwise direction. (d) L reflects each vector 7 about the line x2 = 21 and projects it onto the...
parts a, b, c and d
.
(a) T(31,72) = (2x - 20, -2.61 +5r) on V =R? (b) T(31,12,13) = (-11 + 12,562, 46, -212 +503) on V =R3. (c) T(21, 12) = (2x1 + ix2, 21 +222) on V = C. d] on V = M2x2(R) (with the Frobenius Inner Product). с (d) T ] 3. For each linear operator T in Exercise 1 find an ON basis 3 for V consisting of eigenvectors of T (if possible).
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 , and y3(z) = eェamong its solutions. (b) Now find a different linear homogeneous differential equation of an order lower than the one in (a) that has the same y1,U2,U3 among its solutions. (c) Find a constant coefficient linear homogeneous differential equation of lowest order that...
In Exercise, find the eigenvalues of each linear operator and determine a basis for each eigenspace. T -6x1 - 5x2 + 5x37 - 12 L-10x1 - 10x2 + 9x3]
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