I can already find the components of any vector in a vector space with respect to some basis by finding the coefficients in front of each basis vector. However I've seen some people define the components of a vector in terms of the inner product of the basis vectors and the vector you're trying to decompose.
These two approaches are equal, so why would you use the inner product approach if the basis coefficient approach requires less assumptions? (It isn't dependent on the presence of an inner product).
Best Answer
Because the vector may not be written out explicitly in a component-wise format. Or, it might be the case that we'd like to project our vector $\vec{v}=a\hat{x}+ b\hat{y}$ into a different (e.g. rotated or skewed coordinate system). In this case, we must write out the new basis vectors in terms of the old, and take the inner product. We do not have the luxury of reading off the components in all cases.