Definition of Linear Independence

<aside> 💡 A set of vectors $v_1,v_2,....,v_n$ from a vector space $V$ is said to be linearly independent if $v_1,v_2,....,v_n$ are not linearly dependent.

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Equivalently:

<aside> 💡 A set of vectors $v_1,v_2,...,v_n$ from a vector space $V$ is said to be linearly independent, if the equation: $a_1v_1+a_2v_2+....+a_nv_n=0$ can only be satisfied when $a_i=0$ for all $i=1,2,..,n$

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Equivalently:

<aside> 💡 A set of vectors $v_1,v_2,...,v_n$ from a vector space $V$ is said to be linearly independent if the only linear combination of $v_1,v_2,...,v_n$ which equals 0 is the linear combination with all coefficients 0.

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The 0 vector

• Let $v_1,v_2,...,v_n$ be a set of vectors containing the 0 vector.
• Suppose $v_i=0$. Then we can choose $a_i=1$ and $a_j=0$ for $i\neq j.$
• Then the linear combination $a_1v_1+a_2v_2+....+a_nv_n$ is 0 but not all coefficients are 0.

<aside> 💡 Hence, a set of vectors $v_1,v_2,....,v_n$ containing the 0 vector is always a linearly dependent set.

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When are two non-zero vectors linearly independent?

Linear Independence of Three vectors