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Consider for example the transformation that takes the first basis vector into the average of the first and second and the second basis vector into their difference.(first minus second, say). Its first column would have (1/2)'s in the first two places and 0's elsewhere, the second column will have 1 in the first place and -1 in its second place, again with 0's everywhere else. What will this matrix's transformation do to a vector whose first two coordinates
are a and b and the rest 0? It will produce the vector whose first two coordinates
are In two dimensions, the matrix corresponding to this transformation has columns (1/2, 1/2) and (1, -1) so it looks like:
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respectively, with the rest of its components 0.