Can transformation coefficients be interchanged in symmetric tensors?

[spacetimedude]

Homework Statement

The lecture notes states that if $T_{ij}=T_{ji}$ (symmetric tensor) in frame S, then $T'_{ij}=T'_{ji}$ in frame S'. The proof is shown as $$T'_{ij}=l_{ip}l_{jq}T_{pq}=l_{iq}l_{jp}T_{qp}=l_{jp}l_{iq}T_{pq}=T'_{ji}$$ where relabeling of p<->q was used in the second equality. Where I am confused is that after the 3rd equality, the order of $l_{iq}$ and $l_{jp}$ changes. Is this allowed in all tensors?
Thanks in advance

 

[Orodruin]

The $l_{jp}$ are just numbers so your question boils down to "is $xy = yx$?" where $x$ and $y$ are real numbers.

 

[jedishrfu]

Since the tensor is symmetric then you can interchange the i and j in the expression.

Right?

 

[Orodruin]

Since the tensor is symmetric then you can interchange the i and j in the expression.

Right?

The idea was to show that a tensor with symmetric components in one coordinate system has symmetric components in all coordinate systems. Above this meand that the exchange of p and q in $T_{pq}$ is assumed to be fine and we want to show that this implies that i and j can be exchanged. The OP's question was regarding the validity of changing the order of the transformation coefficients.