By Garrett P.

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See [ 60 ] for a complete justification. 23. (a) We can individually reduce A and B to upper triangular forms U1 and U2 with the determinants equal to the products of their respective diagonal entries. elementary row operations to D will reduce it to the upper triangular form U1 O , and its determinant is equal to the product of its diagonal entries, which O U2 are the diagonal entries of both U1 and U2 , so det D = det U1 det U2 = det A det B. (b) The same argument as in part (a) proves the result.

Unit for Scalar Multiplication: 1 (v, w) = (1 v, 1 w) = (v, w). 14. Here V = C0 while W = R, and so the indicated pairs belong to the Cartesian product vector space C0 × R. The zero element is the pair 0 = (0, 0) where the first 0 denotes the identically zero function, while the second 0 denotes the real number zero. The laws of vector addition and scalar multiplication are (f (x), a) + (g(x), b) = (f (x) + g(x), a + b), c (f (x), a) = (c f (x), c a). 1. e = (x e, y e, z e )T also satisfies x e−y e + 4z e = 0, (a) If v = ( x, y, z )T satisfies x − y + 4 z = 0 and v T e = (x + x e, y + y e, z + z e ) since (x + x e ) − (y + y e) + 4 (z + z e) = (x − y + 4 z) + so does v + v T e −y e +4 z e) = 0, as does c v = ( c x, c y, c z ) since (c x)−(c y)+4 (c z) = c (x−y +4 z) = 0.

We identify each sample value with the matrix entry mij = f (i h, j k). In this way, every sampled function corresponds to a uniquely determined m × n matrix and conversely. Addition of sample functions, (f + g)(i h, j k) = f (i h, j k) + g(i h, j k) corresponds to matrix addition, mij + nij , while scalar multiplication of sample functions, c f (i h, j k), corresponds to scalar multiplication of matrices, c mij . 10. a + b = (a1 + b1 , a2 + b2 , a3 + b3 , . . ), c a = (c a1 , c a2 , c a3 , . .

### Modular curves, raindrops through kaleidoscopes by Garrett P.

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