Determine whether the given vectors are linearly dependent or linearly independent: (i) [0, 1, 1, 2], [3, 1, 5, 2], [−2, 1, 0, 1], [1, 0, 3, −1] (ii) [1, 1, 2], [0, 2, -1], [1, 2,... Determine whether the given vectors are linearly dependent or linearly independent: (i) [0, 1, 1, 2], [3, 1, 5, 2], [−2, 1, 0, 1], [1, 0, 3, −1] (ii) [1, 1, 2], [0, 2, -1], [1, 2, 4] (iii) [1, 2, 1, 0], [2, 3, 0, -1], [1, 2, 1, -1]
Understand the Problem
The question is asking to evaluate a set of vectors to determine if they are linearly dependent or independent. This requires checking if there exists a non-trivial linear combination of the vectors that equals zero.
Answer
The vectors are linearly independent if each column of the RREF of the associated matrix has a pivot; otherwise, they are dependent.
Answer for screen readers
The vectors are independent if each column of the RREF has a pivot; otherwise, they are dependent.
Steps to Solve
- Set Up the Equation
To determine if the vectors are linearly independent, we need to set up the equation:
$$ c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + \ldots + c_n \mathbf{v_n} = \mathbf{0} $$
where $c_1, c_2, \ldots, c_n$ are the scalars and $\mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_n}$ are the vectors.
- Write the Matrix
Transform the vectors into a matrix. Put the vectors as columns in a matrix $A$. For instance, if you have vectors $\mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_n}$, construct:
$$ A = \begin{bmatrix} | & | & \ldots & | \ \mathbf{v_1} & \mathbf{v_2} & \ldots & \mathbf{v_n} \ | & | & \ldots & | \end{bmatrix} $$
- Find the Reduced Row Echelon Form (RREF)
Use row reduction (Gaussian elimination) to bring the matrix $A$ to its Reduced Row Echelon Form (RREF). This will help you analyze the pivot positions.
- Analyze the RREF for Pivots
Examine the RREF matrix you obtained. If every column has a leading 1 (pivot), then the vectors are linearly independent. If any column does not have a leading 1, they are linearly dependent.
- Conclusion
Based on the analysis from the RREF, state whether the vectors are linearly independent or dependent.
The vectors are independent if each column of the RREF has a pivot; otherwise, they are dependent.
More Information
Linear dependence indicates that at least one of the vectors can be written as a linear combination of the others. This concept is fundamental in linear algebra, especially in the study of vector spaces.
Tips
- Not Row Reducing: Failing to reduce the matrix can lead to incorrect conclusions about linear dependence.
- Misinterpreting Pivots: Confusing the presence of free variables with linear independence—free variables indicate dependency.
- Ignoring Zero Vectors: Including the zero vector as a part of the vectors set will always lead to linear dependence.
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