GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 3

Answer 1

Given vectors
$v_{1}$ = $\begin{bmatrix}\ 1\ \\ 0\\ -2\end{bmatrix}$

$v_{2}$ = $\begin{bmatrix}\ -1\ \\0 \\2\end{bmatrix}$

$v_{3}$ = $\begin{bmatrix}\ 1\ \\ 0 \\2 \end{bmatrix}$

The only dependent pair are {$v_{1}$, $v_{2}$} as $v_{2}$ = (-1).$v_{1}$

As $v_{2}$ can be expressed as the linear combination of $v_{1}$ and $v_{3}$. So the given set of vectors are Linearly dependent ($v_{2}$ = (-1).$v_{1}$ + (0).$v_{3}$)

Answer is C

Answer 2

Let’s analyze option by option …

Given set of vectors :

$v_{1} = \begin{bmatrix} 1\\0\\-2 \end{bmatrix}, v_{2} = \begin{bmatrix} -1\\0\\2 \end{bmatrix}, v_{3} = \begin{bmatrix} 1\\0\\2 \end{bmatrix}$

• A. The three vectors $v_{1}$,$v_{2}$ and $v_{3}$ are linearly independent. 

It is FALSE, we can see that $v_{2}$ is linear combination of $v_{3}$ and $v_{1}$, 

                                                                      $v_{2}$ = (-1).$v_{1}$ + 0.$v_{3}$ 

So, this set is linearly dependent.

• A. Every pair of vectors {$v_{1},v_{2}$}, {$v_{2},v_{3}$}, {$v_{1},v_{3}$} are linearly independent. 

 {$v_{1},v_{2}$} is linearly dependent as they are multiple of each other.

 {$v_{2},v_{3}$} is linear independent as they are not multiple of each other.

 {$v_{1},v_{3}$} is linearly independent as they are not multiple of each other.

Statement is saying all pairs are LI. But {$v_{1},v_{2}$} is LD. That’s why These statement is FALSE.

100. All pairs except the pair {$v_{1},v_{2}$} are linearly independent.

This is TRUE as we can see that they are multiple of each other and they are Linearly Dependent ….

                                                                        $v_{2} = (-1).v_{1}$

Other than these all pair are LI. 

500. All pairs except the pair {$v_{1},v_{3}$} are linearly independent.

This is FALSE as we had seen in option (C) {$v_{1},v_{2}$} is LD.

{$v_{1},v_{3}$} is LI as they are not multiple of each other. 

Answer is: C 

Comments

yeah but as you mentioned that 3 elements won't form a pair.so we don't need to consider that {v1,v2,v3}.And if it's been considered then as {v1,v2} are LD so their superset {v1,v2,v3} will also be linearly dependent.

---

@Prashant bhardwaj Even for m=n also we can't directly say whether the given set is linearly dependent or linearly independent  because if n LI vectors are there then it's LI or else it's LD. It's all about vectors set.

---

So, if their superset i.e, {v1,v2,v3} is also LD then option C is also incorrect.

Answer 3

Another approach to look at this question:- the 2^nd part in all the 3d-vectors is 0 i.e., [1 0 -2], [-1 0 2], [1 0 2], so we can look these vectors as a 2d-vectors i.e., [1 -2], [-1 2], [1 2]. (I know these are not equal, but we can look at this approach because any linear combination of these 3d-vectors will always have 2^nd part as 0)

Now we know that, any set of three 2d-vectors can’t be LI. And also [1 -2] = (-1) * [-1 2]. So, Ans is C.

Comments

ultimate visualization!!

---

This answer is totally in a different dimension

Answer 4

v1=[1 0 -2]   v2=[-1 0 2]  v3=[1 0 2]

• A. Linearly dependency or independency is the property of set of vectors not of individual vector.So option A is wrong .
• B. In {v1,v2} [1 0 -2]=(-1)[-1 0 2], in {v2,v3} any of the vector cannot be written as the linear combination of the other vector and also in {v1,v3} the same case . so option B is wrong .
• C. {v2,v3} linearly independent,{v1,v3}  linearly independent,but (v1,v2} is linearly dependent  .so option C is correct.
• D. {v1,v3} linearly independent ,so option d is wrong .