Practice question · Put in order
Order the steps of testing whether three vectors in space are linearly independent.
- Write the three vectors as the columns of a matrix
- Row reduce the matrix to echelon form
- Set up the equation c1 v1 + c2 v2 + c3 v3 = 0
- If every column contains a pivot, only the trivial solution exists and the vectors are independent
Hints
- Independence is a statement about a homogeneous system, so build that system first.
- The verdict is read off the pivot pattern, which only exists after reduction.
Show the answer
- Set up the equation c1 v1 + c2 v2 + c3 v3 = 0
- Write the three vectors as the columns of a matrix
- Row reduce the matrix to echelon form
- If every column contains a pivot, only the trivial solution exists and the vectors are independent
Why
A pivot in every column means no free variables, so the only solution is the trivial one and the vectors are independent. A free variable produces a nontrivial relation and therefore exhibits the redundancy explicitly.
Practise Linear Independence
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