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 \(S_1 = S_2\)?

Answer 1

I know that
\[\left|\matrix{1&0\\0&1\\}\right| - \left|\matrix{0&1\\1&0\\}\right| + \left|\matrix{0&0\\1&0\\}\right| = \left|\matrix{1&-1\\0&1\\}\right|\]

Answer 2

Thing is, do I get more (or less) with \(S_2\) than I get with \(S_1\)?