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Vectoreverything

The journey to tensors begins with scalars and vectors.

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Transpose Matrix

Perhaps the best way to explain it is to start with an example:

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

A^T = \begin{bmatrix} 1 &  4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}

We can express this with an equation of components:

Let A be a matrix and let B be the transpose of A:

a_{ij} = b_{ji}

We give one example:

  • a_{23} = 6
  • b_{32} = 6

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