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Let $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation and let $T$ be a convex subset of $\mathbb{R}^{m}$ . Prove that the set $S=\left\{\mathbf{x} \in \mathbb{R}^{n} : f(\mathbf{x}) \in T\right\}$ is a convex subset of $\mathbb{R}^{n}$ .

since $T$ is convex, point $( 1 - t ) y _ { 1 } + t y _ { 2 }$ must be in $T ,$ lence $f ^ { - 1 } ( 11 -$ $\left. t ) y _ { 1 } + t y _ { 2 } \right) = ( 1 - t ) x _ { 1 } + t x _ { 2 }$ in $S$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 3

Convex Combinations

Vectors

Campbell University

Harvey Mudd College

Baylor University

Idaho State University

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