Consider a two-dimensional spatial coordinate system S' whose coordinates (u,v) are defined by
x = u + v
y = u - v
in terms of the coordinates of a Cartesian coordinate system S. Suppose you are given a vector in S whose contravariant components are Am = (2,8). Determine the contravariant components of this vector in S'.© BrainMass Inc. brainmass.com October 24, 2018, 8:41 pm ad1c9bdddf
You'll find different notations for tensors in the literature. The so-called kernel-index notation makes it particularly easy to remember the transformation rules. In this notation you denote the components of a tensor in a transformed coordinate system (S') by putting a prime on the index, instead of using a different name for the tensor itself. You also do this for the coordinates.
A detailed solution is given. A two-dimensional spatial coordinate systems is defined. The contravariant components of this vector is determined.
Linear Transformations, Matrices, Projections, Reflections, Kernels, Vector Space, Rank and Nullity (11 Problems)
Please see the attached file for the fully formatted problems.
1 Prove that the solution space of AX = 0, where A is a m x n matrix, is a
2 Are the vectors x3 - 1, x2 - x and x linearly independent in P3 ? Why ?
3 Determine whether or not the function T : Mmn --> Mmn dened by T(A) = A + B, where B is a mixed m x n matrix, is a linear transformation. If it is a linear transformation, verify this fact.
4 The function T : R2 --> R2 such that T[(x; y)] = (��x; y) is called a reflection
in the y-axis. Is this function a linear transformation ?
5 Find the kernel and the range for the linear transformation T : R2 ! R2
given by T[(x; y)] = (2x; x �� y)
6 Let the linear transformation T : Mnn --> Mnn be defined by T(A) = A+At.
7 Find the change-of-basis matrix from B to B0 where
B = f(3;-2); (6; 8)g and B0 = f(1; 0); (0; 1)g
8 Let the linear transformation T : R2 ! R2 be defined by T[(x; y)] = (2x + y; x - y). Find [T]BB where B = f(2;-3); (4; 5)g and B0 = fe1; e2g
9 Find the dimension of the solution space of the following homogenous system of linear equations;
x + y - 3x = 0
4x + y + 5z = 0
2x + y + 6z = 0
10 Find the rank and nullity of the linear transformation T : R2 ! R2 given by T(u) = Projv = u where v = (2;��4).
11 Find the 2x2 matrix that describes the following mapping in R2; scaling by 6 in the x-direction and by -8 in the y-direction.
12 Prove that A is similar to A for every n x n matrix A.View Full Posting Details