Discussion week 4

Suppose \(U\) and \(V\) are vector spaces, and let \(Z : U \rightarrow V\) be defined by \(Z(u) = 0_V\) for every \(u \in U\). We want to prove that \(Z\) is a (stupid) linear transformation.

For any \(u_1, u_2 \in U\), we have

\[ Z(u_1 + u_2) = 0_V \]

\[ Z(u_1) + Z(u_2) = 0_V + 0_V = 0_V \]

\(Z(u_1 + u_2) = Z(u_1) + Z(u_2)\)

For any \(u \in U\) and any scalar \(c\), we have

\[ Z(cu) = 0_V \]

And,

\[ c \cdot Z(u) = c \cdot 0_V = 0_V \]

\(Z(cu) = c \cdot Z(u)\)

It’s stupid because multiplying by 0 yields 0.