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(2) Express the quadratic form Q=x; -x} - 4x,x2 + 4xzxz in terms of diagonal quadtratic...
Find an orthogonal change of variables that eliminates the cross product terms in the quadratic form Q, and express Q in terms of the new variables. 7x{ + 6x2 + 5x3 - 4X1X2 + 4x2X3 A substitution x = Py that eliminates cross-product terms is X1 = o A substitution x = Py that eliminates cross-product terms is Xi = – -}y.+3y2– žys, x2 = - - Žy2+3y2 +3v3, x3 = - {yı+ {y2– žv3. The new quadratic form is...
4. Express the quadratic form f(x. x3, xz) = x2 – 2x, x2 + 5x/X3 – x2 + 3x2x3 +3x3? In the form of xTAx. a) A is an upper triangular matrix. b) A is a symmetric matrix.
Consider the quadratic form Q(x) xỈ + x2 + x + 4X1X2 + 4x2x3 + 4x3x1. (a) Find the real symmetric matrix A so that Q(X) = XTAX. (b) Find an orthogonal matrix Q so that the change of variables x = Qy transforms the quadratic for Q(x) into one with no cross-product terms, that is, diagonalize the quadratic form (x). Give the transformed quadratic form. (c) Find a vector x of length 1 at which Q(x) is maximized. (d)...
17) Given the quadratic function g(x)= -x2 - 4x + 4 a) Express the quadratic function in standard forn b) Find its vertex.
5. For the quadratic function f(x) = x2 – - ***2 – 4x + 1: a) write it in the form f(x) = a(x – h)2 + k b) identify the coordinates of the vertex c) state whether there is a maximum or minimum and identify where it occurs
diagonal form for the quadratic form Which of the following is a diagonal form for the quadratic form f = 3x2 – 2xy + 3y2 ? a) u² – 2x² Ob) 2u? – V2 Oc) 4u² + 2v2 O d) 2u? – 4v2
Express the system of differential equations in matrix notation x – 4x + y - (cos t)x = 0 y"+y" - t?x' + 3y'+e-2x = 0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? OA. Xi =X, X2 = X". X3 = y, Xa =y" O B. *= x, X2 = x', *3 = y, X4 =y', X5 =y" OC. *1 =...
A quadratic function f is given. flx) = -x2. 4x 3 (a) Express f in transformation form. f(x) (b) Find the vertex and x and y-intercepts of f. (If an answer does not exist, enter DNE.) (x, y) vertex (x, y) (smaller x-value) x-intercepts (x, y) (larger x-value) (x, y) yintercept (c) Sketch a graph of f. Tools Actions Clear Al 1. Select an object from the Tools menu to the left. Delato 2. Enter coordinates in Object Properties below,...
Quadratic forms arise often in geometry, physics and engineering and it is desirable to reduce these to canonical (standard) forms. Reduce the quadratic form:Q = 3x12 + 5 x22 + 3x32 - 2x2 x3 + 2x1 x3 - 2 x1 x2 to a standard form by making an approximate change in variables X = MY , where M is an orthogonal matrix. i) Write Q as X T AX . ii) Find the Eigenvalues of A. Is the quadratic form...
Convert the following quadratic to standard form by completing the square: x2 + 4x + 12y + 4 = 0.