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Let $S =\{v_1,...,v_k\}$ be a subset of $T=\{v_1,...,v_m\}$. show if $S$ is linearly dependent then $T$ is linearly dependent.

I know to show that something is linearly dependent you should be able to express at least one vector as a linear combination of the others. I don't know how to prove this though. My thought are if I prove $S$ is linearly dependent then I can say that since each vector in $S$ is also in $T$ means that $T$ is also linearly dependent.

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If $S$ is a linearly dependent set, and $S$ is a subset of $T$, then $T$ is linearly dependent. If a vector in $S$ can be represented by a linear combination of other vectors in $S$, then that same vector can be represented by the same linear combination in $T$.

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  • $\begingroup$ Thank you. The combination of the answers really clears things up $\endgroup$ – Susan-l3p Mar 8 '17 at 20:56
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Note that you do not prove that $S$ is linearly dependent. That is already given! So you already know that some vector in $S$ is a linear combination of other vectors in $S$, for example you could write $$c_1v_1+c_2v_2+\cdots+c_kv_k=v_i$$ for some $i$ such that $1\leq i \leq k$. Now, can we use this to show that $v_i$ is also a linear combination of the vectors in $T$?

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  • $\begingroup$ Ooh ok I get it. Thank you $\endgroup$ – Susan-l3p Mar 8 '17 at 20:53
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Your reasoning is correct! Say there's a vector $\boldsymbol{u} \in S$ such that $\boldsymbol{u}$ is a linear combination of vectors in $S$:

$$ \implies \exists \boldsymbol{v_1},\boldsymbol{v_2},\boldsymbol{v_3}\ldots \in S\\ \exists \lambda_1 , \lambda_2 , \lambda_3 \ldots $$ Such that: $$ \boldsymbol{u} = \lambda_1 \boldsymbol{v_1} + \lambda_2 \boldsymbol{v_2} \dots $$

Since $S \subset T$, then $\boldsymbol{u} \in T$, and $\boldsymbol{v_k} \in T$.

This shows that any vector $\boldsymbol{u}$ in $S$, which is a linear combination of vectors in $S$, is also a linear combination of vectors in $T$. And therefore, $T$ is linearly dependent.

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Let's call $s_n$ an ordinary vector of $S$. If $s_n$ can be written as a linear combination of, for example, $s_k$ and $s_p$ (also vectors of $S$), then there is linear dependence in $S$, as stated. Note now that $s_n,s_k$ and $s_p$ also belong to $T$, since $S$ is a subset of $T$. Therefore, there is linear dependence in $T$.

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