# Dimension, Linear Dependence

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Please provide a semi-detailed response for these *two* questions (attached).

In each of the following exercises, we shall use the notation f(x) to denote the function x→f(x), x ε R. In each case, V will denote the vector space of all real-valued functions on the real line, with the vector operations defined point-wise. In each case, determine whether or not the given set is linearly independent and compute the dimension of the linear subspace spanned by the set.

18. {1, eax, ebx}, a≠b.

23. {1, cos2x, sin2x}.

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##### Solution Summary

This solution is comprised of a detailed explanation to determine whether or not the given set is linearly independent and compute the dimension of the linear subspace spanned by the set.

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Problem #18

Suppose holds for all real number . Here . Let and we get . Without the loss the generality, assume . Now let . If , then we get . Then ...

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