Practice question · Sort into groups
Let W be the span of (1, 0, 1) and (0, 1, 1). Decide for each vector whether it lies in W.
Groups: In the span · Not in the span
- (7, -7, 0)
- (0, 0, 0)
- (1, 1, 3)
- (-1, 4, 3)
- (5, 5, 5)
- (2, 3, 5)
Hints
- A general combination is a(1, 0, 1) + b(0, 1, 1) = (a, b, a + b).
- So a vector is in W exactly when its third coordinate is the sum of the first two.
Show the answer
In the span: (2, 3, 5), (0, 0, 0), (-1, 4, 3), (7, -7, 0)
Not in the span: (1, 1, 3), (5, 5, 5)
Why
The span consists of the vectors (a, b, a + b), so membership is the single test 'third coordinate equals first plus second'. Checking: 2 + 3 = 5 yes, 1 + 1 = 2 not 3, -1 + 4 = 3 yes, 5 + 5 = 10 not 5, 7 - 7 = 0 yes. The zero vector always qualifies.
Practise Span and Linear Combinations
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