Practice question · Put in order
Order the steps that decide whether a vector w lies in the span of v1 and v2.
- Write w = c1 v1 + c2 v2 with the scalars c1 and c2 unknown
- Row reduce the system
- Turn the vector equation into a linear system, one equation per coordinate
- If the system is consistent, w is in the span, and the solution gives the coefficients
Hints
- The unknowns are the coefficients, not the vectors.
- Membership of a span is a solvability question, not a geometry question.
Show the answer
- Write w = c1 v1 + c2 v2 with the scalars c1 and c2 unknown
- Turn the vector equation into a linear system, one equation per coordinate
- Row reduce the system
- If the system is consistent, w is in the span, and the solution gives the coefficients
Why
Every span question becomes a linear system in the coefficients, so the whole Gaussian-elimination machinery applies. Consistency means membership; inconsistency means w points somewhere the given vectors cannot reach.
Practise Span and Linear Combinations
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