A linear map sends a position vector to

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

Invariance, Ch Index notation and Cartesian tensors

Problem

A linear map sends a position vector xix_i to yi=Aijxjy_i = A_{ij}x_j, with A=(120010003),x=(1,0,2) m.A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \qquad x = (1, 0, 2)\ \mathrm{m}. ((a)) Identify which index in AijxjA_{ij}x_j is free and which is dummy, and state how many equations yi=Aijxjy_i = A_{ij}x_j represents once the free index is allowed to take each of its values. ((b)) Compute yiy_i, and compute xixix_ix_i and yiyiy_iy_i.

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