Practice question · Multiple choice
The solutions of a linear differential equation form a vector space. Why is noticing that worth anything to someone trying to solve one?
Hints
- If two functions solve the equation, what about their sum? Their scalar multiples?
- Ask what 'basis' means once the objects are solutions rather than arrows.
Show the answer
B. Because every theorem about vector spaces immediately applies
Why
Closure under sums and scaling means solutions have a basis, so a second-order equation needs exactly two independent ones and everything else follows as a combination. The whole solution method is a basis argument, which is the payoff for defining vector spaces abstractly instead of as arrows.
Practise Vector Spaces
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