Question

If vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_{3}and\overrightarrow{v_4}$are linearly independent, then vectors ${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3$ must be linearly independent as well.

Short answer

The above statement is true.

If vectors${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3\text{and}\overrightarrow{v_4}$ are linearly independent, then vectors${\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3$ must be linearly independent as well.

Step-by-step solution

Definition of linearly independent vectors

Let V be a vector space over a field K. Let $v_1,v_2,v_3,...,v_n\in V$, then if there exist scalars$a_1,a_2,a_3,...,a_n\in K$ , 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.

Given the statement

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

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

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

Then eq (1) can be written as

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

To prove whether the vectors v→1,v→2,v→3 are linearly dependent or linearly independent

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

$a_1{\overrightarrow{v}}_1+a_2{\overrightarrow{v}}_2+a_3{\overrightarrow{v}}_3=0$ … (2)

Then eq (2) can be written as

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

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

Thus, our supposition is wrong.

Final Answer

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

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