True or False? If False, you should come up with a counterexample. If True, can you give a proof?
- 1.
- \(T : \RR ^2 \to \RR ^2\), given by \(T(x, y) = (x, -y)\), is a linear transformation.
True False
- 2.
- \(T : \RR ^n \to \RR \), given by \(T(\vec {x}) = \vec {x} \cdot \vec {z}\) for some fixed vector \(\vec {z} \in \RR ^n\), is a linear transformation.
True False
- 3.
- \(T : \RR \to \RR \), given by \(T(x) = x^2\), is a linear transformation.
True False
- 4.
- Let \(T : \RR ^n \to \RR ^m\) be a linear transformation and let \(\vec {v}_{1}, \dots , \vec {v}_{k}\) denote vectors in \(\RR ^n\). If \(\{T(\vec {v}_{1}), \dots , T(\vec {v}_{k})\}\) is linearly independent, then \(\{\vec {v}_{1}, \dots , \vec {v}_{k}\}\) is also linearly
independent.
True False
- 5.
- Let \(T : \RR ^2 \to \RR ^2\) be a linear transformation and suppose \(\vec {v}_{1}, \vec {v}_{2}\) denote vectors in \(\RR ^2\). If \(\{\vec {v}_{1}, \vec {v}_{2}\}\) is linearly independent, then \(\{T(\vec {v}_{1}), T(\vec {v}_{2})\}\) is also linearly
independent.
True False