Question

Let \[ S=\{(2,5),\ (3,7),\ (4,-9),\ (-8,3)\}. \] Then:

• A. \(S\) is linearly independent
• B. \(S\) is linearly dependent
• C. \(S - \{(4, -9)\}\) is linearly independent
• D. \(S - \{(-8, 3)\}\) is linearly independent

Hint:

To solve vector dependency questions quickly, count the number of vectors and compare it to their dimension.
If there are more vectors than dimensions, the set is always linearly dependent.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A set of vectors in a vector space is linearly dependent if at least one of the vectors can be written as a linear combination of the others.
In any vector space of dimension \(d\), the maximum number of linearly independent vectors is equal to \(d\).
Any set containing more than \(d\) vectors in that space is guaranteed to be linearly dependent.

Step 3: Detailed Explanation:
Let us examine the given set of vectors:
\[ S = \{ (2, 5), (3, 7), (4, -9), (-8, 3) \} \]
Each vector in this set has two components, meaning they are elements of the two-dimensional real vector space \(\mathbb{R}^2\):
\[ v_1 = (2, 5), \quad v_2 = (3, 7), \quad v_3 = (4, -9), \quad v_4 = (-8, 3) \]
The dimension of the vector space \(\mathbb{R}^2\) is \(d = 2\).
According to a fundamental theorem of linear algebra:
"Any set of \(m\) vectors in a vector space of dimension \(n\) is linearly dependent if \(m > n\)."
Here, the number of vectors in \(S\) is \(m = 4\), and the dimension of the space is \(n = 2\).
Since \(4 > 2\), the set \(S\) must be linearly dependent.
Let us also evaluate the other options:
- For Option (C): The set \(S - \{(4, -9)\}\) contains 3 vectors: \(\{ (2,5), (3,7), (-8,3) \}\).
Since \(3 > 2\), this reduced set is still linearly dependent.
- For Option (D): The set \(S - \{(-8, 3)\}\) also contains 3 vectors and is therefore linearly dependent.
Thus, the only true statement is that \(S\) is linearly dependent.

Step 4: Final Answer:
Therefore, the correct option is (B).