The OpenGL Viewing Pipeline
... The viewing pipeline refers to the actions necessary to process the specified geometric primitives and show them on the display device. For 2D geometry this is easy. First, we specify the world coordinate window, a region of the 2D plane in which we have defined the geometry. Next, we define the vie ...
... The viewing pipeline refers to the actions necessary to process the specified geometric primitives and show them on the display device. For 2D geometry this is easy. First, we specify the world coordinate window, a region of the 2D plane in which we have defined the geometry. Next, we define the vie ...
Day
... Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is a nonzero if and only if the matrix has a multiplicative inverse Instruction: Discussion & Group Practice Differe ...
... Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is a nonzero if and only if the matrix has a multiplicative inverse Instruction: Discussion & Group Practice Differe ...
Solutions to HW 5
... Exercise 2.4.15: Let V and W be n-dimensional vector spaces, and let T : V → W be a linear transformation. Suppose that β is a basis for V . Prove that T is an isomorphism if and only if T(β) is a basis for W. Proof. We first prove the “only if” implication. So assume that T : V → W is an isomorphi ...
... Exercise 2.4.15: Let V and W be n-dimensional vector spaces, and let T : V → W be a linear transformation. Suppose that β is a basis for V . Prove that T is an isomorphism if and only if T(β) is a basis for W. Proof. We first prove the “only if” implication. So assume that T : V → W is an isomorphi ...
338 ACTIVITY 2:
... If the coefficient matrix is A, we can represent the variables in a (column) vector X and the right-side values of the equations by a (column) vector b and write the system as AX = b. If A has an inverse B (A has to be square and nonsingular) then we can solkve this by multiplying both sides by B to ...
... If the coefficient matrix is A, we can represent the variables in a (column) vector X and the right-side values of the equations by a (column) vector b and write the system as AX = b. If A has an inverse B (A has to be square and nonsingular) then we can solkve this by multiplying both sides by B to ...
Solutions to Homework 2
... 2. Prove that the set of functions β = {1, cos x, cos2 x, . . . , cos6 x} is linearly independent in C(R). (Hint: Suppose c0 · 1 + c1 · cos x + · · · + c6 · cos6 x = 0. Then this equation holds for all values of x. Hence, for each value you substitute in for x, you get a different linear equation in ...
... 2. Prove that the set of functions β = {1, cos x, cos2 x, . . . , cos6 x} is linearly independent in C(R). (Hint: Suppose c0 · 1 + c1 · cos x + · · · + c6 · cos6 x = 0. Then this equation holds for all values of x. Hence, for each value you substitute in for x, you get a different linear equation in ...
Math 2270 - Lecture 33 : Positive Definite Matrices
... A matrix is positive definite if it’s symmetric and all its eigenvalues are positive. The thing is, there are a lot of other equivalent ways to define a positive definite matrix. One equivalent definition can be derived using the fact that for a symmetric matrix the signs of the pivots are the signs ...
... A matrix is positive definite if it’s symmetric and all its eigenvalues are positive. The thing is, there are a lot of other equivalent ways to define a positive definite matrix. One equivalent definition can be derived using the fact that for a symmetric matrix the signs of the pivots are the signs ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.