True or False? If False, you should come up with a counterexample. If True, can you give a proof?
- 1.
- If \(V\) is a subspace of \(\RR ^n\) and \(\vec {x}+\vec {y}\) is in \(V\), then \(\vec {x}\) is in \(V\) or \(\vec {y}\) is in \(V\).
True False
- 2.
- If \(V\) is a set in \(\RR ^n\) such that \(c_1{\vec {v}_1}+c_2{\vec {v}_2}\) is in \(V\) whenever \(\vec {v}_1\) and \(\vec {v}_2\) are in \(V\) for any scalars \(c_1\), \(c_2\), then \(V\) is a subspace.
True False
- 3.
- Every set of four non-zero vectors in \(\RR ^4\) is a basis.
True False
- 4.
- \(\RR ^3\) has a basis of the form \(\left \{\vec {x},\vec {x}+\vec {y},\vec {y}\right \}\).
True False