Final answer:
The statement is false because the performance of algorithms also depends on factors such as constant factors and input size, not just their O(n) notation. A quadratic algorithm can be faster than a linear algorithm for small enough inputs.
Step-by-step explanation:
The statement "For all possible inputs, a linear algorithm to solve a problem must perform faster than a quadratic algorithm to solve the same problem" is false. While it's generally true that a linear algorithm (O(n)) scales better and will eventually be faster as the size of the input (n) grows large, the performance for small inputs or for a specific input size must be empirically tested. Indeed, a quadratic algorithm (O(n2)) could be faster for small inputs due to lower constant factors or more efficient use of resources like caching and memory access patterns.
For example, if we have a linear algorithm with a high constant factor and a quadratic algorithm with a very low constant factor, there could be a range of input sizes for which the quadratic algorithm outperforms the linear one. Only when the input grows large enough will the linear algorithm's growth rate advantage become apparent, and it will surpass the quadratic algorithm in terms of performance.