A linear map preserves sums, scalar multiples, zero and all finite linear combinations. Its kernel and range are subspaces. Its values on a basis determine it uniquely. If $V$ is finite-dimensional, rank plus nullity equals $\dim V$. Translations by nonzero vectors fail linearity.
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- A linear transformation maps 0 to 0.For linear $T$, $T(0)=T(0+0)=T(0)+T(0)$, so subtracting gives $T(0)=0$. This necessary condition provides a quick...
- A linear transformation of a linear combination is the linear combination of the linear transformationIf $T$ is linear, $T(\sum i c iv i)=\sum i c iT(v i)$. Prove this by repeated additivity and homogeneity. This...
- A linear transformation is determined by its action on a basis.If two linear maps agree on a basis, they agree everywhere: expand $v=\sum jc jb j$ and apply linearity to both....
- Definition of how the action of a linear transformation on a basis extends to the whole spaceGiven a basis $B=(b 1,\ldots,b n)$ and chosen vectors $w j\in W$, define $T(\sum jc jb j)=\sum jc jw j$. Unique basis...
- The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.The solution set of $Ax=0$ is $T A^{-1}(\{0\})=\ker T A$. It is a subspace of the domain. For $Ax=b$ with one...
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