general relativity - How to prove a symmetric tensor is indeed a tensor?

Our professor defined a rank $(k,l)$ tensor as something that transforms like a tensor as follows:

$$T^{\mu_1' \mu_2'...\mu_k'}{}_{\nu_1'\nu_2'...\nu_l'} ~=~ \Lambda^{\mu_1'}{}_{\mu_1}...\Lambda^{\mu_k'}{}_{\mu_k}~\Lambda^{\nu_1}{}_{\nu_1'}...\Lambda^{\nu_l}{}_{\nu_l'}~T^{\mu_1\mu_2...\mu_k}{}_{\nu_1\nu_2...\nu_l}$$

Where $\Lambda$ are the Lorentz transformation matrices (translations, rotations, or boosts). I'm not sure if this is only for SR or if also for GR since we've only been talking about SR thus far, though GR is something we'll be covering soon.

He wrote on the whiteboard: if $S_{\mu\nu\rho}=S_{\nu\mu\rho}$ then $S$ is symmetric in $\mu$ and $\nu$.

But let's just talk about a rank (2,0) symmetric contravariant tensor for a second, denoted $S^{\mu\nu}$ and equals $S^{\nu\mu}$. How would we prove this is a tensor? Our book uses $R$ in place of $\Lambda$ in their formulations above, where $R$ might just be rotations. I'm sure General Tensors would have any jacobian and inverse jacobians are matrices rather than just the Lorentz transformations. This is question in Prof. Zee's, "Einstein's Gravity in a Nutshell", Chapter I.4 Exercises 2.

Also if you want to give a student like myself who is new to tensor some advice on learning tensors, and some tensor properties, and how to work with them, be my guest :)

Also, are all transformations homogeneous linear transformations? - These can be read about at: http://www.math.ucla.edu/~baker/149.1.02w/handouts/e_htls.pdf

http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-6/rank-two-tensors/ This lecture gives a nice matrix form of what a symmetric (2,0) tensor looks like. I think this may help, thinking of these tensors as matrices themselves, visually. Basically they're symmetric matrices of the form $A^T=A$. Also we can think of $\Lambda$ if it's a rotation matrix having the property $\Lambda^{-1}=\Lambda$.

Does this picture below do anything for the problem?