Not everything is (only) abstract nonsense

Prove that \mathrm{SL}_2(\mathbb{Z})\to\mathrm{SL}_2(\mathbb{Z}/n\mathbb{Z}) is surjective for every integer n but \mathrm{GL}_2(\mathbb{Z})\to\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z}) is not necessarily surjective.

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Not everything is (only) abstract nonsense

An exercise in Linear Algebra

Let k be a field of characteristic 0 and k[x_1,\ldots,x_n]^{(d)} be the k-vector space of homogeneous degree d polynomials in n variables. Show that the linear k span of the set of d-th powers of linear polynomials (a_1 x_1+\ldots+a_n x_n)^d is the whole space of homogeneous degree d polynomials.

In coordinate-free terms: if V is a finite-dimensional k-vector space then \{ v^n : v\in V\} spans \mathrm{Sym}^dV.

An exercise in Linear Algebra