Image of Linear Combination of Subsets of Vector Space under Linear Transformation

Theorem

Let $K$ be a field.

Let $X$ and $Y$ be vector spaces over $K$.

Let $T : X \to Y$ be a linear transformation.

Let $\lambda, \mu \in K$.

Let $A, B \subseteq X$.

Then:

$T \sqbrk {\lambda A + \mu B} = \lambda T \sqbrk A + \mu T \sqbrk B$

We have:

$\ds T \sqbrk {\bigcup_{x \in A} \paren {\lambda x + \mu B} }$

$=$

$\ds \bigcup_{x \in A} T \sqbrk {\lambda x + \mu B}$

$\ds $

$=$

$\ds \bigcup_{x \in A} \paren {\lambda T x + T \sqbrk {\mu B} }$

$\ds $

$=$

$\ds \bigcup_{x \in A} \paren {\lambda T x} + T \sqbrk {\mu B}$

$\ds $

$=$

$\ds \lambda \bigcup_{x \in A} \paren {T x} + T \sqbrk {\mu B}$

$\ds $

$=$

$\ds \lambda T \sqbrk {\bigcup_{x \in A} x} + T \sqbrk {\mu B}$

$\ds $

$=$

$\ds \lambda T \sqbrk A + T \sqbrk {\mu B}$

$\ds $

$=$

$\ds \lambda T \sqbrk A + \mu T \sqbrk B$

$\blacksquare$