Exercise 1.5.16

Question

Prove that a set of vectors $S$ is linearly independent if and only if any finite subset of $S$ is linearly independent.

Jephian C.-H. Lin · Accepted Answer

Sufficiency: We can prove it by contrapositive statement. If $S$ is linearly dependent we can find $a_1u_1+a_2u_2+\cdots +a_nu_n=0$. But thus the finite set $\{u_1,u_2,\ldots ,u_n\}$ would be a finite subset of $S$ and it's linearly dependent. Necessary: This is the Threorem 1.6.

Exercise 1.5.16

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Exercise 1.5.16

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