Linear Algebra - Subspaces?
True or False and explain your answer.
The set of all 2x2 matrices
[ a b ]
[ c d ]
where bc = 0 is a vector subspace of M.
I know that to prove it a subspace it must be able to hold under addition and scalar multiplication. But thats all I know.
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And be non-empty, I believe. If it is a subspace, then it will contain the identity, so proving that the identity has the desired property is a good way to show non-emptiness.
Now check closure under scalar multiplication and addition. If you multiply by a scalar, do you maintain the desired property? Clearly yes, (bc = 0) => (kb * kc = 0).
Now is there closure under addition? Say you add the matrix:
[ a' b' ]
[ c' d' ].
Does (bc = 0 and b'c' = 0) => ([b + b']*[c + c'] = 0)?