EE 550 Lecture no. 8
... Field : A field consists of a set, denoted by , of elements called scalars and two operations called addition or + and multiplication or . with the operations defined according to the following axioms: ...
... Field : A field consists of a set, denoted by , of elements called scalars and two operations called addition or + and multiplication or . with the operations defined according to the following axioms: ...
Eigenvalues, eigenvectors, and eigenspaces of linear operators
... square matrices under change of basis. Recall that if A and B represent the transformation with respect to two different bases, then A and B are conjugate matrices, that is, B = P −1 AP where P is the transition matrix between the two bases. The eigenvalues are numbers, and they’ll be the same for A ...
... square matrices under change of basis. Recall that if A and B represent the transformation with respect to two different bases, then A and B are conjugate matrices, that is, B = P −1 AP where P is the transition matrix between the two bases. The eigenvalues are numbers, and they’ll be the same for A ...
570 SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A
... 4. The Discriminant Matrix of an Algebraic Equation. We shall now consider two applications of the foregoing theory to the case of the commutative algebra generated by an algebraic equation. Let ...
... 4. The Discriminant Matrix of an Algebraic Equation. We shall now consider two applications of the foregoing theory to the case of the commutative algebra generated by an algebraic equation. Let ...
linear transformations and matrices
... Definition 3.5 (Range). Let T : V → W be a linear transformation. The range of T , denoted R(T ), is defined as R(T ) := {T (v) : v ∈ V }. Remark 3.6. Note that R(T ) is a subspace of W , so its dimension can be defined. Definition 3.7 (Rank). Let V, W be vector spaces over a field F. Let T : V → W ...
... Definition 3.5 (Range). Let T : V → W be a linear transformation. The range of T , denoted R(T ), is defined as R(T ) := {T (v) : v ∈ V }. Remark 3.6. Note that R(T ) is a subspace of W , so its dimension can be defined. Definition 3.7 (Rank). Let V, W be vector spaces over a field F. Let T : V → W ...
Lie Groups, Lie Algebras and the Exponential Map
... it associates to X a one-parameter subgroup of G (homomorphism from R to G) and recalling that the exponential map is precisely the map identifying elements of g with such homomorphisms. Recall the following basic facts about groups that are subgroups of GL(n, R), these will be the ones we will most ...
... it associates to X a one-parameter subgroup of G (homomorphism from R to G) and recalling that the exponential map is precisely the map identifying elements of g with such homomorphisms. Recall the following basic facts about groups that are subgroups of GL(n, R), these will be the ones we will most ...
Summary of week 8 (Lectures 22, 23 and 24) This week we
... of A (for all i and j). An n × n complex matrix is said to be unitary if t AA = I. Calculations similar to those above show that A is unitary if and only if x 7→ Ax is an isometry of Cn , which in turn is equivalent to the columns of A forming an orthonormal basis of Cn . See 5.15 of [VST]. A real v ...
... of A (for all i and j). An n × n complex matrix is said to be unitary if t AA = I. Calculations similar to those above show that A is unitary if and only if x 7→ Ax is an isometry of Cn , which in turn is equivalent to the columns of A forming an orthonormal basis of Cn . See 5.15 of [VST]. A real v ...
Lines and Planes
... which is called "u perp." What is the magnitude of u? ? Why are u and u? both in the plane with normal n? What is the angle between u and u? ? 40. Suppose that u is orthogonal to a unit vector n and that u? = n u: Show that u = cos ( ) u + sin ( ) u? is a vector in the plane spanned by u and u? that ...
... which is called "u perp." What is the magnitude of u? ? Why are u and u? both in the plane with normal n? What is the angle between u and u? ? 40. Suppose that u is orthogonal to a unit vector n and that u? = n u: Show that u = cos ( ) u + sin ( ) u? is a vector in the plane spanned by u and u? that ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.