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Suppose that $\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$ is an affinely independent set in $\mathbb{R}^{n}$ and $\mathbf{q}$ is an arbitrary point in $\mathbb{R}^{n} .$ Show that the translated set $\left\{\mathbf{p}_{1}+\mathbf{q}, \mathbf{p}_{2}+\mathbf{q}, \mathbf{p}_{3}+\mathbf{q}\right\}$ is also affinely independent.

See explanation.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 2

Affine Independence

Vectors

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so we'll prove this by contradictions. So improved these bye contradiction. And what I mean by contradiction is we suppose that's p one plus Que Pete's ooh plus Que Pete during Plus que is are finally dependence, right, because we want to show that it should be finally Independence. But let's assume that it is a fine independent there. We should arrive at the contradiction now. This would imply that's just exist. There is some non zero skill laws c. One seat to sit, Cherie Such duds c one plus seeds to plus Secretary records zero on dhe would have duds c one times p one plus que rights Lost seats. Sue Time Speed Super Askew Lost City Every time. Speed three plus Que record zero Doesn't definition off something off this sets being are finally dependent. That's always just don't we just used the definition. No one will expand these. You're gonna have C one p one lost. See one Q plus. It's soupy, too. Plus, it's a cue loss Sitara Petri plus Citterio Cube record, too. Zero If you bring this C one pizza getter, you have C one p one lost seats. Soupy, too. Loss seat threepeat three Now Q is common to all of these. So you have a cute year. You have acute here. You have a cute here so I can pull the cue outs. Double blow us. See one plus two plus three Q equal to zero. Well, we have that C one policy to plus it, Terry. It cost zero. So this would just be C one p one loss. It's who. So this isn't go to zero. It's C to pizza. Ooh, plus seat during Peak. Terry in court soon. Zero. That's what you have left. Now What does this mean under recall Dots C. One plus it's who plus it very equal to zero on We gods we got from the previous speech. Don't see one p one plus seats. Soupy TSU plus Secretary Petteri equal to zero. This just implies that's the sets P want pizza p three. These are finally dependence, right? This is the definition off. I find I find dependence because they're getting a C one people. Plus it's a pizza. Plus, it's a great victory. Cost. Zero. Given that C one plus, it's two plus it very close to zero. This is Sean's C one seats to sit very are non zero now. This gives a contradiction because in our hypothesis is recalled over it all. That's supposed to be one a pizza. Peter is finally independence, so this gives a contradiction. One soon, a lot. I put This is what you're given now recall. That's the reason why we have this contradiction is because we already as Judah this set was finally dependence. So because we not around us, a contradiction this would imply that's p one plus que pizza. Ooh, plus cube, Pete three plus que These are finally independence independence.

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