Solutions to Math 51 Final Exam — June 8, 2012
... • The main issue was forgetting to give a basis, and instead giving the span of the basis. • A few students computed N (A) instead of N (A − 3I). (b) Determine the definiteness of Q. Justify your answer. (4 points) Since Q(e1 ) = −1 and Q(e3 ) = 2, Q assumes both positive and negative values and hen ...
... • The main issue was forgetting to give a basis, and instead giving the span of the basis. • A few students computed N (A) instead of N (A − 3I). (b) Determine the definiteness of Q. Justify your answer. (4 points) Since Q(e1 ) = −1 and Q(e3 ) = 2, Q assumes both positive and negative values and hen ...
Lesson Plan -
... that an equation such as y 5 3x 1 5 is a prescription for determining a second number when a first number is given. Gr. 5 AF 1.0: Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results. Also included: Gr ...
... that an equation such as y 5 3x 1 5 is a prescription for determining a second number when a first number is given. Gr. 5 AF 1.0: Students use variables in simple expressions, compute the value of the expression for specific values of the variable, and plot and interpret the results. Also included: Gr ...
Lines and Planes
... Part 1: Equations of Lines In this section, we use vectors, the cross product, and the dot product to explore lines and planes in 3-dimensional space. We begin by exploring the vector equation of a line in the xy-plane. To begin with, a position vector is a vector P = hx; yi whose initial point is … ...
... Part 1: Equations of Lines In this section, we use vectors, the cross product, and the dot product to explore lines and planes in 3-dimensional space. We begin by exploring the vector equation of a line in the xy-plane. To begin with, a position vector is a vector P = hx; yi whose initial point is … ...
Chapter 1 PLANE CURVES
... If p = (x0 , x1 , x2 ) is a point of P2 , and if x0 6= 0, we may normalize the first entry to 1 without changing the point: (x0 , x1 , x2 ) ∼ (1, u1 , u2 ), where ui = xi /x0 . We did this for P1 above. The representative vector (1, u1 , u2 ) is uniquely determined by p, so points with x0 6= 0 corre ...
... If p = (x0 , x1 , x2 ) is a point of P2 , and if x0 6= 0, we may normalize the first entry to 1 without changing the point: (x0 , x1 , x2 ) ∼ (1, u1 , u2 ), where ui = xi /x0 . We did this for P1 above. The representative vector (1, u1 , u2 ) is uniquely determined by p, so points with x0 6= 0 corre ...
Exact Differential Equation
... Equations reducible to The Exact Equations :Sometimes a differential equation which is not exact may become so, on multiplication by a suitable function known as the integrating factor. ...
... Equations reducible to The Exact Equations :Sometimes a differential equation which is not exact may become so, on multiplication by a suitable function known as the integrating factor. ...
Discuss on Variation of Parameters
... Because of the In x term, the right‐hand side is not one of the special forms that the method of undetermined coefficients can handle; variation of parameters is required. The first step requires obtaining the general solution of the corresponding homogeneous equation, y″ – 2 y′ + y = 0: ...
... Because of the In x term, the right‐hand side is not one of the special forms that the method of undetermined coefficients can handle; variation of parameters is required. The first step requires obtaining the general solution of the corresponding homogeneous equation, y″ – 2 y′ + y = 0: ...
Homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane.