How do you put vectors into columns to test for linear independence?
Conditions
- You have a set of vectors to test.
- You are forming a homogeneous system .
- The vectors are placed as columns in matrix .
Reasoning, step by step
- Construct a matrix where each column is one of the tested vectors.
- Set up the homogeneous equation , where is the vector of coefficients.
- Perform row reduction (Gaussian elimination) on the matrix.
- Check for free variables in the reduced form.
- If free variables exist, the vectors are linearly dependent.
- If no free variables exist, the vectors are linearly independent.
Example
The script states: 'Write the coefficient equation as a homogeneous matrix system, placing the tested vectors into columns. Row reduction reveals whether free variables remain.'
Common misconceptions
- Placing vectors as rows instead of columns (which tests row independence, but the standard method for column vectors uses columns).
- Forgetting that row reduction is used to solve the homogeneous system, not just to find the rank.
- Assuming that a zero determinant always implies independence (it implies dependence for square matrices).
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