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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-7-18\">Problem 7.18</a> \n>\n> Show that covariant symmetric tensors of rank 2, satisfying $T _ { i j } = T _ { j i }$ , over a vector space $V$ of dimension $n$ form a vector space of dimension $n ( n + 1 ) / 2$\n>\n> (a) A tensor $S$ of type $( 0 , r )$ is called <u class=\"concept-term\" data-concept-identity=\"urn:uuid:09932e00-f8f2-44fe-9842-4c9fe5aa15dd\" data-concept-route=\"/concept-identities#concept-09932e00-f8f2-44fe-9842-4c9fe5aa15dd\" id=\"first-02fa9e816bfba2f0393f-en\">totally symmetric</u> if $S _ { i _ { 1 } i _ { 2 } \\ldots i _ { r } }$ is left unaltered by any interchange of indices. What is the dimension of the vector space spanned by the totally symmetric tensor on $V ?$\n>\n> (b) Find the dimension of the vector space of covariant tensors ofrank 3 having the cyclic symmetry\n>\n> $$\n> T (u, v, w) + T (v, w, u) + T (w, u, v) = 0.\n> $$",
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