Question:
If \( \{ v_1, v_2, v_3 \} \) is a linearly independent set of vectors in a vector space over \( \mathbb{R} \), then which one of the following sets is also linearly independent?
If \( \{ v_1, v_2, v_3 \} \) is a linearly independent set of vectors in a vector space over \( \mathbb{R} \), then which one of the following sets is also linearly independent?
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To check for linear independence, see if any vector can be expressed as a combination of others. If not, the set is independent.
Updated On: Dec 15, 2025
\( \{ v_1 + v_2 - v_3, 2v_1 + v_2 + 3v_3, 5v_1 + 4v_2 \} \)
- \( \{ v_1 - v_2, v_2 - v_3, v_3 - v_1 \} \)
\( \{ v_1 + v_2 - v_3, v_2 + v_3 - v_1, v_3 + v_1 - v_2, v_1 + v_2 + v_3 \} \)
- \( \{ v_1 + v_2, v_2 + 2v_3, v_3 + 3v_1 \} \)
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The Correct Option is D
Solution and Explanation
To determine which of the given sets is linearly independent, we must understand the concept of linear independence. A set of vectors is said to be linearly independent if the only scalar coefficients that satisfy the linear combination equating to zero are all zeroes. That means if we have a set of vectors \(\{ \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n \}\), they are linearly independent if the equation:
\[c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + \ldots + c_n\mathbf{u}_n = \mathbf{0}\]implies \(c_1 = c_2 = \ldots = c_n = 0\). Now, let's analyze the given options:
- \( \{ v_1 + v_2 - v_3, 2v_1 + v_2 + 3v_3, 5v_1 + 4v_2 \} \)
- \( \{ v_1 - v_2, v_2 - v_3, v_3 - v_1 \} \)
- \( \{ v_1 + v_2 - v_3, v_2 + v_3 - v_1, v_3 + v_1 - v_2, v_1 + v_2 + v_3 \} \)
- \( \{ v_1 + v_2, v_2 + 2v_3, v_3 + 3v_1 \} \)
Therefore, the set \( \{ v_1 + v_2, v_2 + 2v_3, v_3 + 3v_1 \} \) is the linearly independent set. Hence, the correct answer is:
\( \{ v_1 + v_2, v_2 + 2v_3, v_3 + 3v_1 \} \)
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