# Advice for proving existence claims

I’m pretty new to proof—based mathematics and so far I’ve noticed that one type of proof I seem to struggle the most with is proving an existence claim. Particularly, showing the existence of some object with a specified property. From what I gather looking at solutions, this process involves defining some object in a clever way that guarantees its existence and allows you to derive the desired property from its definition. But I feel like I have no idea how to carry out that process systematically. Any advice?

For reference here is an example of the kind of problem I’ve struggled with:

Suppose w1…wn is a basis of W and V is finite dimensional. Prove there is a basis v1…vm of V st with respect to these bases, all entries in the first row of M(T) are 0, except for possibly a 1 in the first entry of the first row.

Where for some basis v1…vn of and a basis w1…wm and some linear transformation
T : V → W, M(T) is defined as the matrix consisting of the entries Aj,k equal to the coefficients in T(vk) = A1,k•w1 + … + Am,k•wk

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