Geodesic
An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices
Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 761-768

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Zbl
We provide and discuss an elementary proof of the exponential con- ditioning of real Vandermonde matrices which can be easily given in undergraduate courses: we exclusively use the definition of conditioning and the sup-norm formula on $[-1, 1]$ for Chebyshev polynomials of first kind. The same proof idea works virtually unchanged for the famous Hilbert matrix. 
Serra Capizzano, Stefano. An Elementary Proof of the Exponential Conditioning of Real Vandermonde Matrices. Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 761-768. http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a20/
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     title = {An {Elementary} {Proof} of the {Exponential} {Conditioning} of {Real} {Vandermonde} {Matrices}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {761--768},
     year = {2007},
     volume = {Ser. 8, 10B},
     number = {3},
     zbl = {1139.15002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2007_8_10B_3_a20/}
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