1. FINITE-DIMENSIONAL VECTOR SPACES
... A basis for a finite-dimensional vector space V is a linearly independent set that spans V. Example 16: The set {(1, 0, 0, … , 0, 0), (0, 1, 0, … , 0, 0), … , (0, 0, 0, …, 0, 1)} is a basis for Fn, where F is any field. This is called the standard basis. We often denote it by {e1, e2, … , en}. A min ...
... A basis for a finite-dimensional vector space V is a linearly independent set that spans V. Example 16: The set {(1, 0, 0, … , 0, 0), (0, 1, 0, … , 0, 0), … , (0, 0, 0, …, 0, 1)} is a basis for Fn, where F is any field. This is called the standard basis. We often denote it by {e1, e2, … , en}. A min ...
E4 - KFUPM AISYS
... Be ordered bases for R 3 and R2 , respectively. Determine the linear transformation L : R 3 R 2 , whose representation with respect to S and T is A . ...
... Be ordered bases for R 3 and R2 , respectively. Determine the linear transformation L : R 3 R 2 , whose representation with respect to S and T is A . ...
Jones Vector Treatment of Polarization
... Consider a linear polarizer with transmission axis along the vertical (y). Let a 2X2 matrix represent the polarizer operating on vertically polarized light. The transmitted light must also be vertically polarized. Thus, ...
... Consider a linear polarizer with transmission axis along the vertical (y). Let a 2X2 matrix represent the polarizer operating on vertically polarized light. The transmitted light must also be vertically polarized. Thus, ...
Cartesian tensor
In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is through an orthogonal transformation.The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product.Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity. While orthonormal bases may be found for some such coordinate systems (e.g. tangent to spherical coordinates), Cartesian tensors may provide considerable simplification for applications in which rotations of rectilinear coordinate axes suffice. The transformation is a passive transformation, since the coordinates are changed and not the physical system.