Consider the linear system X' = AX where A is defined by , where a and...
Consider the linear system X' = AX where A is defined by
,
where a and b are real numbers. Assume that the determinant of A is not zero.
Classify the equilibrium solution (0,0) depending on the signs of a and b. Also, sketch a few trajectories for each case, including a few tangent vectors.
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Consider the linear system X' = AX where A is defined by
,
where a and...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and ide...
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), b...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
(6) (Hand in!) Consider the linear function L : R3 → R3, given by the formula L(2)-Ax, where A is...
(6) (Hand in!) Consider the linear function L : R3 → R3, given by the formula L(2)-Ax, where A is one of the following 3x 3 matrices: 0 -1 3 0 -21 0 1-2 3 1 -3 -1 2 0, 1 -1 1 21-3 Compute the determinant of each matrix. In each case what information does the determinant give you regarding: (i.) the invertibility of the function of L, (ii.) the volume of the parallelogram formed by passing the standard...
Consider the linear system of first order differential equations x' = Ax, where x= x(t), t...
Consider the linear system of first order differential equations x' = Ax, where x= x(t), t > 0, and A has the eigenvalues and eigenvectors below. 4 2 11 = -2, V1 = 2 0 3 12 = -3, V2= 13 = -3, V3 = 1 7 2 i) Identify three solutions to the system, xi(t), xz(t), and x3(t). ii) Use a determinant to identify values of t, if any, where X1, X2, and x3 form a fundamental set of...
7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#A...
please explain
everything!
7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#AX with A-|-1 İa] This family can be represented in the trace-determinant plane as a curve. -1 0 a. Sketch the curve representing this family in the trace-determinant plane. We were unable to transcribe this image
7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#AX with A-|-1 İa] This family can be represented in the...
2 x [b] Consider the following linear system of equations AX =B : (i) Determine a...
2 x [b] Consider the following linear system of equations AX =B : (i) Determine a basis for the row space of A. (ii) Compute the Rank of the augmented matrix (A:B), then use it to classify the solution of this system (Unique - Many -No: solution). (iii) Is the matrix A diagonalizable? Explain your answer and verify the similarity transformation.
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the syst...
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
Consider a linear system of the form d/dt x(t) = Ax. Next suppose there exists a...
Consider a linear system of the form d/dt x(t) = Ax. Next suppose there exists a positive definite solution to the matrix inequality: A^TP + PA < 0 where M<0 means M is negative definite. Then prove that all eigenvalues of A have negative real part.
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determina...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determinant of the coefficient matrix c+1 c (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stablhe. (4) c=흘 (2) c=-2 (1) c=-1 (3) c=-8
Consider an autonomous...
Consider the linear system of first order differential equations x' = Ax, where x = x(t),...
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
Consider the system of two coupled differential equations: y-cx + dy, x-ax + by, with the equilibrium solution (xe,ye) = (0,0) (a) Rewrite the coupled system as a matrix differential equation and identify the matrix A. Obtain a general solution to the matrix differential equation in terms of eigenvectors and eigenvalues of A. Justify your answer (b) Classify possible types and stability of the equilibrium with dependence on the eigenvalues of A. (Note: You are not asked to compute the...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
(6) (Hand in!) Consider the linear function L : R3 → R3, given by the formula L(2)-Ax, where A is one of the following 3x 3 matrices: 0 -1 3 0 -21 0 1-2 3 1 -3 -1 2 0, 1 -1 1 21-3 Compute the determinant of each matrix. In each case what information does the determinant give you regarding: (i.) the invertibility of the function of L, (ii.) the volume of the parallelogram formed by passing the standard...
Consider the linear system of first order differential equations x' = Ax, where x= x(t), t > 0, and A has the eigenvalues and eigenvectors below. 4 2 11 = -2, V1 = 2 0 3 12 = -3, V2= 13 = -3, V3 = 1 7 2 i) Identify three solutions to the system, xi(t), xz(t), and x3(t). ii) Use a determinant to identify values of t, if any, where X1, X2, and x3 form a fundamental set of...
please explain
everything!
7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#AX with A-|-1 İa] This family can be represented in the trace-determinant plane as a curve. -1 0 a. Sketch the curve representing this family in the trace-determinant plane. We were unable to transcribe this image
7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#AX with A-|-1 İa] This family can be represented in the...
2 x [b] Consider the following linear system of equations AX =B : (i) Determine a basis for the row space of A. (ii) Compute the Rank of the augmented matrix (A:B), then use it to classify the solution of this system (Unique - Many -No: solution). (iii) Is the matrix A diagonalizable? Explain your answer and verify the similarity transformation.
Consider the plane autonomous system 4) 2 X'=AX with A (a) Find two linearly independent real solutions of the system (b) Classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). (c) Plot the phase portrait of the system containing a trajectory with direction as t-oo whose initial value is X(0) (0,6)7 and any other trajectory with direc- tion. (You do not need to draw solution curves explicitly.)
Consider the plane...
Consider an autonomous system , = (1 + c)x + cy where c is a real constant. (a) Calculate the trace T and the determinant of the coefficient matrix c+1 c (b) For each following cases of c, classify the stability (stable or unstable) and the type (center, node, saddle, or spiral) of the critical point (0,0). Note that if a critical point is a center, it is stablhe. (4) c=흘 (2) c=-2 (1) c=-1 (3) c=-8
Consider an autonomous...
Consider the linear system of first order differential equations x' = Ax, where x = x(t), t > 0, and A has the eigenvalues and eigenvectors below. Sketch the phase portrait. Please label your axes. 11 = 5, V1 = 12 = 2, V2 = ()
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