Practice question · Sort into groups
In the plane R^2, sort each set of vectors by what it does.
Groups: Spans the whole plane · Spans only a line · More than a basis (redundant)
- (2, 2) alone
- (1, 0), (0, 1) and (1, 1)
- (3, 0) alone
- (1, 0) and (0, 1)
Hints
- Two independent vectors span the plane; a single vector spans a line.
- Three vectors in the plane must contain a redundant one.
Show the answer
Spans the whole plane: (1, 0) and (0, 1)
Spans only a line: (2, 2) alone, (3, 0) alone
More than a basis (redundant): (1, 0), (0, 1) and (1, 1)
Why
Two independent vectors (s1) span the plane and form a basis. One vector (s2, s4) spans only the line through it. Three vectors in a 2D space (s3) cannot all be independent, one is redundant.
Practise Linear Independence and Basis
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More questions on Linear Independence and Basis
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