Register

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 9

• edited by
1,020 views
9 votes
9 votes

Given a set of vectors $\text{S}(|\text{S}| \geq n)$, with all vectors in $\mathbb{R}^n$, which of the following is a necessary and sufficient condition for the vectors of $\text{S}$ to be Linearly Dependent?

  1. Exactly $n$ vectors can be represented as a linear combination of other vectors of the set $\text{S}$.
  2. At least $n$ vectors can be represented as a linear combination of other vectors of the set $\text{S}$.
  3. At least one vector $u$ can be represented as a linear combination of any vector(s) of the set $\text{S}$.
  4. At least one vector $u$ can be represented as a linear combination of vectors (other than $u$ ) of the set $\text{S}$.
• edited by
User Avatar

4 Answers

12 votes
12 votes
This follows directly from the definition of Linear Independence. $\text{C}$ can't be the answer because it says "any vector(s)", which would include the vector $u$ itself. $\text{A} \;\&\; \text{B}$ are sufficient, but not necessary.
• edited by
User Avatar
1 votes
1 votes
We know that definitions are always “necessary and sufficient “ statement and it is implicit.So given a set of vectors S(|S|>=n) with all vectors in Rn is said to be linearly dependent if and only if at least one vector can be represented as the linear combination of the others means representing at least one vector of set S as the linear combination of the other vectors is the necessary and sufficient condition for S to be linearly dependent.option A and option B are the sufficient condition not the necessary ,and according to the definition C is neither necessary nor sufficient

And D is the correct option according to the definition.
User Avatar
1 votes
1 votes
Given that the number of vectors in the set S are >=n.

2 cases can ply:

1. |S| (# vectors in set S) > n ------> It would mean that whatever be , as the no. of vectors in S is >n for R^n . It is always going to be linearly dependent.

2.  |S| (# vectors in set S)  =  n ------> If that is the case then we can say that if At least one vector can be represented as a linear combination of vectors (other than itself ) of the set S then we can say it to be Linearly dependent, else Linearly Independent.
User Avatar
Voting & Accepting an answer helps improve the community
Answer:
← PreviousNext →
← Previous in categoryNext in category →

Related questions

970
views
3 answers
10 votes
GO Classes asked Mar 14, 2023
970 views
GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 7
Which of the following set is/are linearly independent?$\left\{\left[\begin{array}{l}1 \\ 2\end{array}\right],\left[\begin{array}{l}2 \\ 1\end{array}\right]\right\}$ ...
User Avatar
1.7k
views
6 answers
23 votes
GO Classes asked Mar 14, 2023
1,684 views
GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 10
Which of the following is(are) sufficient argument(s) to show that the vectors of set $\text{S}$ ...
User Avatar
1.4k
views
5 answers
17 votes
GO Classes asked Mar 14, 2023
1,417 views
GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 11
Which of the following is(are) true for the following system of linear equations $\text{AX}=\overrightarrow{0}$ ... $\text{AX}=\overrightarrow{0}$.
User Avatar
746
views
3 answers
10 votes
GO Classes asked Mar 14, 2023
746 views
GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 12
Given a set $\text{S}$ of vectors in $\mathbb{R}^n$, and set $\text{X(X} \subset \text{S})$ and $\text{Y(S} \subset \text{Y}),$ mark all ... $\text{S}$ are Linearly Independent, then the vectors of $\text{Y}$ are also Linearly Independent.
User Avatar
Link Copied!