Exercise 2.A.1 (Exercise 2.A.1)

Suppose v 1 , v 2 , v 3 , v 4 spans V . Prove that the list

v 1 − v 2 , v 3 − v 3 , v 3 − v 4 , v 4

also spans V .

Answers

For any v , ∃ ⁡ a 1 , a 2 , a 3 , a 4 , such that

v = a 1 v 1 + a 2 v 2 + a 3 v 3 + a 4 v 4 = a 1 ( v 1 − v 2 ) + ( a 1 + a 2 ) ( v 2 − v 3 ) + ( a 1 + a 2 + a 3 ) ( v 3 − v 4 ) + ( a 1 + a 2 + a 3 + a 4 ) v 4 		(1)

So, v 1 − v 2 , v 2 − v 3 , v 3 − v 4 , v 4 spans V .

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2025-03-16 22:11
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V = span ( v 1 , v 2 , v 3 , v 4 ) = span ( v 1 , v 2 , v 3 , v 4 , v 1 − v 2 , v 2 − v 3 , v 3 − v 4 ) ( since  v 1 − v 2 , v 2 − v 3 , v 3 − v 4 ∈ span ( v 1 , v 2 , v 3 , v 4 ) ) = span ( v 4 , v 1 − v 2 , v 2 − v 3 , v 3 − v 4 ) ( since  v 1 , v 2 , v 3 , v 4 ∈ span ( v 1 − v 2 , v 2 − v 3 , v 3 − v 4 ) ) 		(1)

Thus, as desired, v 1 − v 2 , v 2 − v 3 , v 3 − v 4 , v 4 spans V .

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2025-03-16 23:45
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