A basis is changed according to and

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Tensors and differential geometry

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

A basis is changed according to

\mathbf{e}'_1 = 2\mathbf{e}_1

and

\mathbf{e}'_2 = \mathbf{e}_1 + \mathbf{e}_2

, so that

\mathbf{e}'_i = A^j{}_i\mathbf{e}_j

with

A = \begin{pmatrix}2 & 1\\ 0 & 1\end{pmatrix}

(columns indexed by

i

). ((a)) A vector has contravariant components

V^i = (3, 2)

in the original basis. Find its components

V'^i

in the primed basis. ((b)) A gradient has covariant components

W_i = (4, 5)

in the original basis. Find its components

W'_i

in the primed basis.

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