1.13 Linear independence

Definition 1

Likewise if ${\sum }_{j=1}^N{\alpha }_j{v}_j=0$ only when ${\alpha }_j=0$ $\forall j$ , then ${\left\{v_j\right\}}_{j=1}^N$ is said to be linearly independent .

Find ${\alpha }_1,{\alpha }_2,{\alpha }_3$ such that ${\alpha }_1{v}_1+{\alpha }_2{v}_2+{\alpha }_3{v}_3=0$ . [ ${\alpha }_1=1,{\alpha }_2=-3,{\alpha }_3=1$ .] Note that any two vectors are linearly independent.

Note that if a set of vectors ${\left\{v_j\right\}}_{j=1}^N$ are linearly dependent then we can remove vectors from the set without changing the span of the set.