Solution
= Solution
{c}
Every <elementary column operation> preserves the <linear span> of the columns: each new column is a <linear combination> of the old columns, and the inverse operation supplies the reverse inclusion. Thus the <column space> of $A''$ equals that of $A$, which is precisely the <image of a linear map> $T_\alpha$. Consequently \b[(B) is true for every $\alpha\in\mathbb R$].