Liner Inepenene Definition : Te set of vetors: v, v,., v p is Linerly Inepenent if te eqution, v + v +... + p v p s only te trivil solution: k for k,,..p. Tt sme set of vetors is Linerly Depenent if tere exists weigts,,,. p,not ll zero, su tt: v + v + + p v p. Exmple : Fin te vlues of "" for wi te given set of vetors re linerly epenent. u v 5 w Te set is Linerly Inepenent if: u only if. + v + 3 w A ( u v w) A 5 5 3 Pge of 9
3 + 6 U If 3, ten 3 is free vrile, tus mitting n infinity of solutions to te mtrix eqution. Terefore, tere o exist weigts, not ll zero, su tt: u + v + 3 w provie 3 n tus te vetors re Linerly Depenent for tt vlue of "". Liner Inepenene of Mtrix Columns :Te mtrix eqution, A x, n e written s te vetor eqution: x + x +.. + x n n, feturing te olumns of te oeffiient mtrix s te vetors. Te olumns of te "A" mtrix re tus Linerly Inepenent if n only if te trivil solution is te only solution to: Ax. If, owever, te eqution Ax s more tn te trivil solution, ten te olumns of "A" must e Linerly Depenent. If te olumns of "A" re Linerly Depenent, n "A" is squre, ten eterminnt of "A" must e zero. Pge of 9
THEOREM : A set, v, v,., v p, is Linerly Depenent if n only if t lest one of te vetors in te set is liner omintion of te oters. m Suppose v k i v i, were i k pn m < p. i i m i v i Let k. Tus, m v k i v i sine not ll of te i re zero, te i set of vetors {v,..., v m } is Linerly Depenent. But tis set is suset of te set {v,..., v p }, wi must terefore e Linerly Depenent. Exmple : Determine if te given set of vetors is Linerly Depenent in R 3. u v 5 w 3 3 5 + 5 Pge 3 of 9
w 5 u + v Terefore, te set:{ u, v, w} is Linerly Depenent euse t lest one vetor in te set is liner omintion of te oters. Exmple 3: Determine if te set: { Linerly Inepenent in R 3., 4, } is 4 A 4 4 A 4 4 Sine "A" is singulr, its olumns re Linerly Depenent set. Pge 4 of 9
Exmple 4: Determine if te set: {,,, } is Linerly Inepenent in R 3. A Tus, te set {,, } is Linerly Inepenent. α β γ Pge 5 of 9
α β γ Terefore, te Linerly Inepenent set of olumns of "A" spn ll of R 3 euse te vetor is ritrry. Alterntively, sine only 3 vetors re neee to spn ll of R 3, set of 4 vetors must e Linerly Depenent. THEOREM : Let v, v,., v n, e vetors in vetor spe "V". A vetor "x" in Spn{ v, v,., v n } n e written uniquely s linerly omintion of tose vetors if n only if te set: { v, v,., v n } is Linerly Inepenent. Wt tis teorem sys is tt every vetor in te suspe spnne y { v, v,., v n } n e eompose into unique liner omintion of tose vetors only if { v, v,., v n } is Linerly Inepenent set or tt if { v, v,., v n } is Linerly Depenent set, te eomposition is not unique. Pge 6 of 9
Exmple 5: Given te funtions: " x" n " x ", sow tt tese two vetors re linerly inepenent in C[, ] n linerly epenent in C[, ]. If te two funtions re Linerly Depenent, ten for ll vlues of "x", tere exists unique onstnt "α" su tt ( x ) α x. If x, ten x x n α n tus te funtions re Linerly Depenent on C[, ]. In ontrst, if x <, ten x x, n α. Sine "α" is not unique in C[, ], te funtions re Linerly Inepenent in C[, ]. Definition : Te Vetor Spe C ( n ) [, ] is te set of ll funtions wit ontinuous erivtives of orer "( n )". Pge of 9
Definition : Let te set {f, f,, f n } e in te Vetor Spe C ( n ) [, ]. Ten, te Wronskin of {f, f,, f n } on [, ] is enote s W(f, f,, f n ), were: W(f, f,, f n ) f x f x n n f f x f x n n f f n x f n x n n f n THEOREM : Let f, f,., f n, e elements of C ( n ) [, ]. If tere exists t lest one vlue x ε[, ] su tt Wf (, f,., f n ), ten te set:{f, f,, f n } is Linerly Inepenent set of funtions. If te set:{f, f,, f n } were Linerly Depenent, ten te oeffiient mtrix wose eterminnt is tt Wronskin woul e singulr n tus Wf (, f,., f n ) for ll xε[, ].r. Pge 8 of 9
Exmple 6: Sow tt te funtions: "e x " n "x e x " re linerly inepenent in C(, ). ( ) We x, xe x e x e x xe x ( x + ) e x e x xe x e x e x ( ) e x Sine We x, xe x for ll vlues of "x", ten te given pir of funtions re Linerly Inepenent on te intervl (, ). Pge 9 of 9