JEE Main & Advanced Mathematics Vector Algebra Linear Independence and Dependence of Vectors

(1) Linearly independent vectors : A set of non-zero vectors \[{{\mathbf{a}}_{1}},\,{{\mathbf{a}}_{2}},.....{{\mathbf{a}}_{n}}\] is said to be linearly independent, if  \[{{x}_{1}}{{\mathbf{a}}_{1}}+{{x}_{2}}{{\mathbf{a}}_{2}}+.....+{{x}_{n}}{{\mathbf{a}}_{n}}=\mathbf{0}\Rightarrow {{x}_{1}}={{x}_{2}}=.....={{x}_{n}}=0\].

(2) Linearly dependent vectors : A set of vectors \[{{\mathbf{a}}_{1}},\,{{\mathbf{a}}_{2}},.....{{\mathbf{a}}_{n}}\]  is said to be linearly dependent if there exist scalars \[{{x}_{1}},\,{{x}_{2}},......,{{x}_{n}}\] not all zero such that \[{{x}_{1}}{{\mathbf{a}}_{1}}+{{x}_{2}}{{\mathbf{a}}_{2}}+.....+{{x}_{n}}{{\mathbf{a}}_{n}}=\mathbf{0}\]

Three vectors \[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k}\],  \[\mathbf{b}={{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\]  and  \[\mathbf{c}={{c}_{1}}\mathbf{i}+{{c}_{2}}\mathbf{j}+{{c}_{3}}\mathbf{k}\] will be linearly dependent vectors iff \[\left| \,\begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}}  \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}}  \\ \end{matrix}\, \right|\,\,=\,\,0\].

Properties of linearly independent and dependent vectors

(i) Two non-zero, non-collinear vectors are linearly independent.

(ii) Any two collinear vectors are linearly dependent.

(iii) Any three non-coplanar vectors are linearly independent.

(iv) Any three coplanar vectors are linearly dependent.

(v) Any four vectors in 3-dimensional space are linearly dependent.