JHMT 2015 Algebra Test Solutions 14 February 2015 1. In a Super
... Finally, we show 6 6 is achievable. In AM-GM, equality is only achieved when 2 = a so p p p p p p p a = 2 6. If we let x = 2 6 and y = 2 6, then a = 2 6 so 6 6 is achievable. Hence, the p minimum possible value is 6 6 . Solution 2: Let a = x2 + y 2 and b = xy. Then we are trying to minimize 2a + b. ...
... Finally, we show 6 6 is achievable. In AM-GM, equality is only achieved when 2 = a so p p p p p p p a = 2 6. If we let x = 2 6 and y = 2 6, then a = 2 6 so 6 6 is achievable. Hence, the p minimum possible value is 6 6 . Solution 2: Let a = x2 + y 2 and b = xy. Then we are trying to minimize 2a + b. ...
On derivatives of polynomials over finite fields through integration
... A detailed study of the cryptanlytic significance of linear structures was initiated by Evertse [7] in which cryptanysis of DES like ciphers are discussed along with several possible extensions. Linear structures are also considered by Nyberg and Knudsen in a paper on provable security against a dif ...
... A detailed study of the cryptanlytic significance of linear structures was initiated by Evertse [7] in which cryptanysis of DES like ciphers are discussed along with several possible extensions. Linear structures are also considered by Nyberg and Knudsen in a paper on provable security against a dif ...
1 - Mu Alpha Theta
... zero. Since Det(AB) = Det(A)Det(B), statements II and III must be true. If S is a singular matrix, Det(SS) = Det(S)Det(S) = 0*0 = 0. Also, Det(S*A) = Det(S)Det(A) = 0*Det(A) = 0. Statements I and IV can be shown to not be true. For example, the sum of the following two singular matrices is ...
... zero. Since Det(AB) = Det(A)Det(B), statements II and III must be true. If S is a singular matrix, Det(SS) = Det(S)Det(S) = 0*0 = 0. Also, Det(S*A) = Det(S)Det(A) = 0*Det(A) = 0. Statements I and IV can be shown to not be true. For example, the sum of the following two singular matrices is ...
F18PA2 Number Theory and Geometry: Tutorial 9
... (a) Find a direct isometry f mapping (2, 3) 7→ (7, 1) and (4, −1) 7→ (3, 3). Show that f is a rotation, and find its centre. (b) Find an opposite isometry g that does the same thing. Show that g is a glide, and find the equation of its axis. Sketch the effects of these isometries on the reference tr ...
... (a) Find a direct isometry f mapping (2, 3) 7→ (7, 1) and (4, −1) 7→ (3, 3). Show that f is a rotation, and find its centre. (b) Find an opposite isometry g that does the same thing. Show that g is a glide, and find the equation of its axis. Sketch the effects of these isometries on the reference tr ...
Uniqueness of the row reduced echelon form.
... row equivalent iff there is a finite sequence of row operations that transform A into B. Or what is the same thing if there is a finite number of elementary matrices E1 , . . . , Ek is that B = Ek · · · E1 A. This is very closely related to matrix multiplication because of Theorem 2.2 Two m×n matric ...
... row equivalent iff there is a finite sequence of row operations that transform A into B. Or what is the same thing if there is a finite number of elementary matrices E1 , . . . , Ek is that B = Ek · · · E1 A. This is very closely related to matrix multiplication because of Theorem 2.2 Two m×n matric ...
Semidefinite and Second Order Cone Programming Seminar Fall 2012 Lecture 10
... Consider (A, B, ) where is a bilinear operator that is : A × A → B. A and B are finite-dimensional real linear spaces with dim A = m and dim(B) = n. Note that bilinearity assumption is equivalent to the distributive law: • a (αb + βc) = αa b + βa c • (αb + βc) a = αb a + βc a Note a ...
... Consider (A, B, ) where is a bilinear operator that is : A × A → B. A and B are finite-dimensional real linear spaces with dim A = m and dim(B) = n. Note that bilinearity assumption is equivalent to the distributive law: • a (αb + βc) = αa b + βa c • (αb + βc) a = αb a + βc a Note a ...
Equation
... Use the inverse operation for division, which is multiplication. m/3 = 4 m is divided by 3 3*(m/3) = 3*4 Multiply by 3, both sides m = 12 ...
... Use the inverse operation for division, which is multiplication. m/3 = 4 m is divided by 3 3*(m/3) = 3*4 Multiply by 3, both sides m = 12 ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.