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Vector transformation

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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'.

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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.

The ...

Solution Summary

A detailed solution is given. A two-dimensional spatial coordinate systems is defined. The contravariant components of this vector is determined.

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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
vector space.

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 de ned 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.
Find ker(T).

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.

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