Exercise [13.14]

Shaun Culver · 29 May 2008, 15:19

$x^a\to \left( \begin{array}{ccc} T^1{}_1 & T^1{}_2 & T^1{}_3 \\ T^2{}_1 & T^2{}_2 & T^2{}_3 \\ T^3{}_1 & T^3{}_2 & T^3{}_3 \end{array} \right)\left( \begin{array}{c} x^1 \\ x^2 \\ x^3 \end{array} \right)$

The product of two matrices $T^a{}_b (3\times 3)$ and $x^b (3\times 1)$ is defined as

$R^a{}_c=\sum _{b=1}^3 T^a{}_bx^b$

$R^a{}_c=\left(T^a{}_1x^1\right)+\left(T^a{}_2x^2\right)+\left(T^a{}_3x^3\right)$

Which forms a $(3\times 1)$ matrix

$x^a\to \left( \begin{array}{c} \left[T^1{}_1x^1+T^1{}_2x^2+T^1{}_3x^3\right] \\ \left[T^2{}_1x^1+T^2{}_2x^2+T^2{}_3x^3\right] \\ \left[T^3{}_1x^1+T^3{}_2x^2+T^3{}_3x^3\right] \end{array} \right)$

Exercise [13.14]

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