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Let $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a linear transformation, let $T$ be anaffine subset of $\mathbb{R}^{m},$ and let $S=\left\{\mathbf{x} \in \mathbb{R}^{n} : f(\mathbf{x}) \in T\right\} .$ Show that $S$ is an affine subset of $\mathbb{R}^{n}$ .

In conclusion, $f\left((1-t) x_{1}+t x_{2}\right)$ is in $T,$ so $(1-t) x_{1}+t x_{2}$ is in $S$ meaningthat $S$ is affine subset.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 1

Affine Combinations

Vectors

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