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In Exercises 9 and $10,$ mark each statement True or False. Justify each answer. a. If $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}$ is an affinely dependent set in $\mathbb{R}^{n},$ then the set $\left\{\tilde{\mathbf{v}}_{1}, \ldots, \overline{\mathbf{v}}_{p}\right\}$ in $\mathbb{R}^{n+1}$ of homogeneous forms may be linearly independent. b. If $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3},$ and $\mathbf{v}_{4}$ are in $\mathbb{R}^{3}$ and if the set $\left\{\mathbf{v}_{2}-\mathbf{v}_{1}, \mathbf{v}_{3}-\mathbf{v}_{1}, \mathbf{v}_{4}-\mathbf{v}_{1}\right\}$ is linearly independent, then $\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{4}\right\}$ is affinely independent. c. Given $S=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{k}\right\}$ in $\mathbb{R}^{n},$ each $\mathbf{p}$ in aff $S$ has a unique representation as an affine combination of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{k}$ d. When color information is specified at each vertex $\mathbf{v}_{1}, \mathbf{v}_{2}$ $\mathbf{v}_{3}$ of a triangle in $\mathbb{R}^{3},$ then the color may be interpolated at a point $\mathbf{p}$ in aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ using the barycentric coordinates of $\mathbf{p}$ . e. If $T$ is a triangle in $\mathbb{R}^{2}$ and if a point $\mathbf{p}$ is on an edge of the triangle, then the barycentric coordinates of $\mathbf{p}$ (for thistriangle) are not all positive.

F,T,T,T,T

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 2

Affine Independence

Vectors

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so we have some Suren Force question. Question A us to be false. There isn't easy. That's the sets view on ZP in our n Plus one. It's not that it's may be generally independent. It is its most be literally independence number B is true and this follows from certain five C is also true. This is the statements off Syrian six D is true. You can see example five on DPA rdgraf Abovitz on e We're told to um you see the triangle. The answer is true. Some coordinates will be zero.

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