Proving an equation with a triple scalar product and tensor Is there anybody who can help me with this proof? I know the basics, but can't figure this out.
Let $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ be vectors and $\boldsymbol{T}$ a tensor. Recall that the mixed product of the three vectors is denoted by $$\boldsymbol{a}\cdot(\boldsymbol{b} \times \boldsymbol{c}) = [\boldsymbol{a},\boldsymbol{b},\boldsymbol{c}].$$ Prove that $$[\boldsymbol{Ta},\boldsymbol{Tb},\boldsymbol{Tc}] = \det(\boldsymbol{T})[\boldsymbol{a},\boldsymbol{b},\boldsymbol{c}].$$ Provide a geometric interpretation.

How you prove this depends on where you are starting from. In particular, if you know $$\boldsymbol b \times \boldsymbol c = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}=\mathbf i \begin{vmatrix}b_2&b_3\\c_2&c_3\end{vmatrix} + \mathbf j \begin{vmatrix}b_3&b_1\\c_3&c_1\end{vmatrix} + \mathbf k \begin{vmatrix}b_1&b_2\\c_1&c_2\end{vmatrix}$$then it becomes obvious that $$\boldsymbol a \cdot (\boldsymbol b \times \boldsymbol c) = a_1 \begin{vmatrix}b_2&b_3\\c_2&c_3\end{vmatrix} + a_2 \begin{vmatrix}b_3&b_1\\c_3&c_1\end{vmatrix} + a_3 \begin{vmatrix}b_1&b_2\\c_1&c_2\end{vmatrix}=\begin{vmatrix} a_1 & a_2 & a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}$$
Now determinants are unchanged by transposition, so we can also write
$$\boldsymbol a \cdot (\boldsymbol b \times \boldsymbol c) =\begin{vmatrix} a_1 & b_1 & c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix} = \det(\begin{bmatrix}\boldsymbol a&\boldsymbol b&\boldsymbol c\end{bmatrix})$$
(Note that the bracket notation on the right is not the triple product notation you wrote, but rather represents the matrix whose columns are the vectors $\boldsymbol a, \boldsymbol b,$ and $\boldsymbol c$.) 
By the rules of matrix multiplication, $$\boldsymbol T\begin{bmatrix}\boldsymbol a&\boldsymbol b&\boldsymbol c\end{bmatrix} = \begin{bmatrix}\boldsymbol{Ta}&\boldsymbol{Tb}&\boldsymbol{Tc}\end{bmatrix}$$
And since the determinant is multiplicative, $$\begin{align}\boldsymbol{Ta} \cdot (\boldsymbol{Tb} \times \boldsymbol{Tc}) &= \det(\begin{bmatrix}\boldsymbol{Ta}&\boldsymbol{Tb}&\boldsymbol{Tc}\end{bmatrix})\\&=\det(\boldsymbol T\begin{bmatrix}\boldsymbol a&\boldsymbol b&\boldsymbol c\end{bmatrix})\\&=\det(\boldsymbol T)\det(\begin{bmatrix}\boldsymbol a&\boldsymbol b&\boldsymbol c\end{bmatrix})\\&=\det(\boldsymbol T)\boldsymbol (a \cdot (\boldsymbol b \times \boldsymbol c))\end{align}$$
