Question

4. Find four vectors v1,...,V4 in R* that satisfies the condition that {V1, V2, V3} are linearly indepen- dent, but {v1, V2, V3, V,} are linearly dependent.

PLEASE MAKE IT AS CLEAR AS POSSIBLE

Answer 1

$\textbf{4.}$ Consider the vectors ,

V1 = (1,0,0,0)

$v_{2}=(0,1,0,0)$

U3 = (0,0,1,0)

$v_{4}=(1,1,1,0)$

First we will prove that the set  $\left \{ v_{1},v_{2},v_{3} \right \}$ is linearly independent .

Consider the relation  

$\Rightarrow c_{1} (1,0,0,0)+c_{2}(0,1,0,0)+c_{3}(0,0,1,0) =(0,0,0,0)$

$\Rightarrow (c_{1},0,0,0)+(0,c_{2},0,0)+(0,0,c_{3},0) =(0,0,0,0)$

$\Rightarrow (c_{1},c_{2},c_{3},0) =(0,0,0,0)$

Equating each entry both side we get ,

$c_{1}=0,c_{2}=0$ and  $c_{3}=0$

So if then $c_{1}=0,c_{2}=0$ and  $c_{3}=0$ . Hence the   set  $\left \{ v_{1},v_{2},v_{3} \right \}$ is linearly independent .

Now we will prove that the set  $\left \{ v_{1},v_{2},v_{3},v_{4} \right \}$ is linearly dependent .

$\Rightarrow v_{1}+v_{2}+v_{3}=v_{4}$

That is one vectors in the set  $\left \{ v_{1},v_{2},v_{3},v_{4} \right \}$ can be written as linear combination of others

Hence  $\left \{ v_{1},v_{2},v_{3},v_{4} \right \}$ is linearly dependent set .

.

.

.

.

If you have any doubt please comment and don't forget to rate the answer . Your rating keeps us motivated.