Linear Algebra I · Linear maps, kernels, images, and rank-nullity
A composite preserves two-term linear combinations
Problem
Let $K$ be a field, let $U,V,W$ be $K$-vector spaces, and let $S:U\to V$ and $T:V\to W$ be linear. For all $a,b\in K$ and $x,y\in U$, prove $(T\circ S)(ax+by)=a(T\circ S)(x)+b(T\circ S)(y)$.
Hint
Expand $(T\circ S)(ax+by)$ as $T(S(ax+by))$.
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