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

See proof.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 3

Convex Combinations

Vectors

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

02:56

In mathematics, a vector (…

06:36

05:58

Let $f : \mathbb{R}^{n} \r…

04:11

04:12

Let $S$ be an affine subse…

04:51

If $c \in \mathbb{R}$ and …

05:16

Let $\mathbf{v}$ be an ele…

04:16

If $A$ and $B$ are convex …

03:51

Let $\left\{\mathbf{p}_{1}…

02:44

Let $T : \mathbb{R}^{m} \r…

06:54

Let $T$ be a linear transf…

02:40

Suppose vectors $\mathbf{v…

and this exercise, we are going to prove that the image of a comeback set on their linear transformation is also comebacks. So let's take X and Y in FFS so X can be written as if off X hat on Why can be written us f uh, why hot where X had on why had are elements of s then 40 in the close intervals Everyone, we have one minus three times X plus three times Why is the same as one minus t times ffx hat plus three times f of my heart on since official in your transformation, we have that this is the same as if off one minus t time Sex hat plus t times y hot on now sues IHS was a comeback set on x hahd on why hot were elements of s. Then we conclude that one Manistee time sex hard pressed eat them So I had It's also in development affairs on since this number isn't this if off this if off one minus tedium sex had plus stadium. So I had is an element off FFs on this implies that all minus t times explicitly times y is an f off s So ffs is a comeback set as we wanted to prove

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