Question:
Which of the following set is linearly dependent set in \(R^3(R)\)?
Which of the following set is linearly dependent set in \(R^3(R)\)?
Show Hint
Any set of vectors containing the zero vector is automatically linearly dependent.
Updated On: Jul 3, 2026
- \(\{(-1,0,1),(0,1,1),(2,-2,-1)\}\)
- \(\{(0,0,1),(0,1,1),(0,0,0)\}\)
- \(\{(1,0,1),(0,1,1),(1,1,0)\}\)
- \(\{(0,1,2),(0,0,1)\}\)
Show Solution
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The Correct Option is B
Solution and Explanation
Concept:
A set of vectors is linearly dependent if one of the vectors can be written as a linear combination of the others. Also, any set of vectors containing the zero vector is always linearly dependent.
Step 1: Observe option (B). Option (B) is \[ \{(0,0,1),(0,1,1),(0,0,0)\} \] This set contains the zero vector \[ (0,0,0) \]
Step 2: Use the standard result. If a set contains the zero vector, then we can choose a non-zero scalar multiplying the zero vector and still get the zero vector. For example, \[ 1(0,0,0)+0(0,0,1)+0(0,1,1)=(0,0,0) \] Here the coefficients are not all zero. So the set is linearly dependent.
Step 3: Final answer. \[ \boxed{\{(0,0,1),(0,1,1),(0,0,0)\}} \]
A set of vectors is linearly dependent if one of the vectors can be written as a linear combination of the others. Also, any set of vectors containing the zero vector is always linearly dependent.
Step 1: Observe option (B). Option (B) is \[ \{(0,0,1),(0,1,1),(0,0,0)\} \] This set contains the zero vector \[ (0,0,0) \]
Step 2: Use the standard result. If a set contains the zero vector, then we can choose a non-zero scalar multiplying the zero vector and still get the zero vector. For example, \[ 1(0,0,0)+0(0,0,1)+0(0,1,1)=(0,0,0) \] Here the coefficients are not all zero. So the set is linearly dependent.
Step 3: Final answer. \[ \boxed{\{(0,0,1),(0,1,1),(0,0,0)\}} \]
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