In Linear Algebra Done Right, it defines the linear maps of a homogeneous system of linear equations with n variables and m equations as T(x1,…,xn)=(∑k=1nA1,kxk,…,∑k=1nAm,kxk)=0 My question is why the homogeneous system of linear equations can be expressed as the linear maps from Fn↦Fm

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In Linear Algebra Done Right, it defines the linear maps of a homogeneous system of linear equations with n variables and m equations as  T(x1,…,xn)=(∑k=1nA1,kxk,…,∑k=1nAm,kxk)=0  My question is why the homogeneous system of linear equations can be expressed as the linear maps from Fn↦Fm

Yosef Krause · Accepted Answer

A homogeneous system of linear equations is, by definition, of the form    A11x1+A12x2+⋯+A1nxn=0    Am1x1+Am2x2+⋯+Amnxn=0    where all Aij  are fixed scalars, i.e. are ∈F(which is typically R or Q for the first round).    Now, consider the linear map T:BFn↦BFm defined by the given formula, i.e. a tuple (x1,⋯,xn) of scalars is mapped to the tuple (column vector) given by the left hand sides of the equations, that is,    T = (x1,⋯,xn)↦(∑kA1kxk,⋯,∑kAmkxk)    You can readily verify that this indeed defines a linear map, and that the set of equations is exactly its kernel    kerT={x∈BFn:T(x)=0}.    Moreover, and most importantly, if we put the coefficients in a matrix A, and regard tuples as column vectors, then we simply have    ∀x∈BFn: T(x)=A⋅x

In Linear Algebra Done Right, it defines the linear maps

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