Solution Key
... (6) (15 Points) Let S : R → R be a linear map with nullity(S) = 2. Then show directly (that is without using the rank plus nullity theorem) that rank(S) = 3. Solution: As dim ker(S) = nullity(S) = 2 the subspace ker(S) of V has a basis v1 , v2 with two elments. The vector space R5 is 5 dimensional t ...
... (6) (15 Points) Let S : R → R be a linear map with nullity(S) = 2. Then show directly (that is without using the rank plus nullity theorem) that rank(S) = 3. Solution: As dim ker(S) = nullity(S) = 2 the subspace ker(S) of V has a basis v1 , v2 with two elments. The vector space R5 is 5 dimensional t ...
Section 13.1 Vectors in the Plane
... X-1, 2\ = [- , 1_ and - u = - X-1, 2\ = [ , -1_ have half the length of u and are parallel to u. ...
... X-1, 2\ = [- , 1_ and - u = - X-1, 2\ = [ , -1_ have half the length of u and are parallel to u. ...
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.