Chapter 4 - Vector Spaces - 4.7 Change of Basis - Problems - Page 319: 38

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Let $(v_1,...v_n)$ be a base for $B$ Obtain $x=a_1v_1+a_2v_2+...+a_nv_n, f\forall x\in B, c \in C$ then $[x]_B=\begin{bmatrix} a_1\\ a_1 \\ . \\ . \\ a_n \end{bmatrix}$ Thus, we have $[cx]_B=\begin{bmatrix} ca_1\\ ca_1 \\ . \\ . \\ ca_n \end{bmatrix}=c\begin{bmatrix} a_1\\ a_1 \\ . \\ . \\ a_n \end{bmatrix}=c[x]_B$ Hence, the property is true.