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0 $begingroup$ Let $A$ be a linearly independent subset of a subspace $S$ . If $xnotin S$ , show that $Acup{x}$ is linearly independent. Theorem : Let $B$ be linearly independent and $ynotin B$ . Then, $Bcup{y}$ is linearly dependent iff $yin Span(B)$ . Using this theorem, I get that- $A$ is linearly independent and $xnotin SRightarrow xnotin A$ . So, $Acup{x}$ is linearly independent iff $xnotin Sp(A)$ . Therefore, it should be enough to show that- $Asubseteq S$ (where, $A$ is linearly independent) and $xnotin S$ $Rightarrow xnotin Sp(A)$ $_{...(1)}$ $Rightarrow Acup{x}$ is linearly independent $_{...(2)}$ My question is, if my approach is correct how should I prove $(1)$ because $(2)$ obviously follows from there. Or is my approach incorrect? [There are similar questions on the site, bu...