Question

Suppose
S_1 = \rm span\left\{ \left|\matrix{1&0\\0&1\\}\right|, \left|\matrix{0&1\\1&0\\}\right|, \left|\matrix{1&-1\\0&1\\}\right| \right\}
and
S_2 = \rm span\left\{ \left|\matrix{1&0\\0&1\\}\right|, \left|\matrix{0&1\\1&0\\}\right|, \left|\matrix{0&0\\1&0\\}\right| \right\}

Does \(S_1 = S_2\)?