What do nonzero collinear vectors and an all-zero set span?
Conditions
- For the first case, the vectors are nonzero and collinear.
- For the second case, all vectors in the set are the zero vector.
Reasoning, step by step
- Identify the vectors as nonzero and collinear.
- Recognize that their linear combinations are restricted to a single line.
- Conclude that they span a line.
- Identify the vectors as an all-zero set.
- Recognize that any linear combination of zero vectors is the zero vector.
- Conclude that the set spans only the origin.
Example
Nonzero collinear vectors span a line, and an all-zero set spans only the origin.
Common misconceptions
- Assuming collinear vectors can span a 2D plane.
- Believing an all-zero set spans a line or a plane.
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