91影视

Let V be a vector space over a field K. Let $v_1,v_2,v_3,...,v_n鈭�V$, then if there exist scalars$a_1,a_2,a_3,...,a_n鈭�K$ , not all of them 0, such that

$a_1{v}_1+a_2{v}_2+...a_n{v}_n=0$

Then the vectors$v_1,v_2,v_3,...,v_n$ are called linearly dependent vectors.

Otherwise, linearly independent vectors.

Since it is given that the vectors${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3\text{and}\stackrel{鈫�}{v_4}$ are linearly independent then there exist scalars $a_1,a_2,a_3,\text{and}{a}_4$, all of them 0, such that

$a_1{\stackrel{鈫�}{v}}_1+a_2{\stackrel{鈫�}{v}}_2+a_3{\stackrel{鈫�}{v}}_3+a_4{\stackrel{鈫�}{v}}_4=0........\left(1\right)$

Since,$a_1=0=a_2=a_3=a_4$

Then eq (1) can be written as

$a_1{\stackrel{鈫�}{v}}_1+a_2{\stackrel{鈫�}{v}}_2+a_3{\stackrel{鈫�}{v}}_3+0.{\stackrel{鈫�}{v}}_4=0$

Let us suppose${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3$ are linearly dependent vectors, then there exist scalars$a_1,a_2,\text{and}{a}_3$ , not all of them zero, such that

$a_1{\stackrel{鈫�}{v}}_1+a_2{\stackrel{鈫�}{v}}_2+a_3{\stackrel{鈫�}{v}}_3=0$ 鈥� (2)

Then eq (2) can be written as

$a_1{\stackrel{鈫�}{v}}_1+a_2{\stackrel{鈫�}{v}}_2+a_3{\stackrel{鈫�}{v}}_3+0.{\stackrel{鈫�}{v}}_4=0$

If $a_1鈮�0,a_2鈮�0,\text{and}{a}_3鈮�0$then this will contradict the fact that ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3\text{and}\stackrel{鈫�}{v_4}$are linearly independent vectors.

Thus, our supposition is wrong.

If vectors${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3\text{and}\stackrel{鈫�}{v_4}$are linearly independent, then we found scalars $a_1,a_2,\text{and}{a}_3$such that the vectors ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3$are linearly independent.

Hence, if vectors are ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3\text{and}\stackrel{鈫�}{v_4}$linearly independent, then vectors ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2,{\stackrel{鈫�}{v}}_3$must be linearly independent as well.