Linearly Independent Sets and Linearly
... These functions are “vectors” in the vector space P 2 . Is the set of vectors p 1 , p 2 , p 3 linearly independent or linearly dependent? If this set is linearly dependent, then give a linear dependence relation for the set. Solution We need to consider the vector equation c1p1 c2p2 c3p3 ...
... These functions are “vectors” in the vector space P 2 . Is the set of vectors p 1 , p 2 , p 3 linearly independent or linearly dependent? If this set is linearly dependent, then give a linear dependence relation for the set. Solution We need to consider the vector equation c1p1 c2p2 c3p3 ...
8.1 General Linear Transformation
... If V is a finite-dimensional vector space and T:V ->V is a linear operator then the following are equivalent. (a)T is one to one (b) ker(T) = {0} (c)nullity(T) = 0 (d)The range of T is V;that is ,R(T) =V ...
... If V is a finite-dimensional vector space and T:V ->V is a linear operator then the following are equivalent. (a)T is one to one (b) ker(T) = {0} (c)nullity(T) = 0 (d)The range of T is V;that is ,R(T) =V ...
2 Session Two - Complex Numbers and Vectors
... Vectors may be expressed in Cartesian coordinates (with unit vectors î, ĵ, k̂ pointing along the x, y, z axes) or in various forms of polar coordinates, e.g. r̂, θ̂, φ̂. Note that î and ĵ here have nothing directly to do with imaginary numbers, and that some authors use different conventions for ...
... Vectors may be expressed in Cartesian coordinates (with unit vectors î, ĵ, k̂ pointing along the x, y, z axes) or in various forms of polar coordinates, e.g. r̂, θ̂, φ̂. Note that î and ĵ here have nothing directly to do with imaginary numbers, and that some authors use different conventions for ...
Lines and planes
... Writing a = OA and r = OP , we have that P lies on the line if and only if r = a + λv This is called the vector equation of the line. Two lines are parallel if and only if their direction vectors are parallel or opposite; that is the lines r = a + λv and r = b + µw are parallel if and only if v = kw ...
... Writing a = OA and r = OP , we have that P lies on the line if and only if r = a + λv This is called the vector equation of the line. Two lines are parallel if and only if their direction vectors are parallel or opposite; that is the lines r = a + λv and r = b + µw are parallel if and only if v = kw ...
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.