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In Exercises 11 and $12,$ mark each statement True or False. Justify each answer.a. The set of all affine combinations of points in a set $S$ is called the affine hull of $S$ .b. If $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{k}\right\}$ is a linearly independent subset of $\mathbb{R}^{n}$ and if $\mathbf{p}$ is a linear combination of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{k},$ then $\mathbf{p}$ is an affine combination of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{k}$c. The affine hull of two distinct points is called a line.d. A flat is a subspace.e. A plane in $\mathbb{R}^{3}$ is a hyperplane.

a) trueb) falsec) trued) truee) true

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 1

Affine Combinations

Vectors

Dylan C.

November 28, 2021

The typed answer for part d should be false. It seems that there was a mistype.

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and it's exercise a statement. A is true. In fact, what we have here is the definition off their fine hall on this can be verified by going to page for 139. A statement V is false on. We can consider the following counter example in our to consider the vector who isn't resort to to This is a linear combination, but not on a fine combination. Off the victor 11 statement see is true. When we have two vectors there, I find combinations Have the following form on this is just the equation of a life. So statement see is true, then statement d is false. I remember that Ah Flat is a translation of a safe space in this, uh, growing w is a flat and we can see that the victory zero is not contained in W Therefore, w cannot be a super space sort of statement. Thes falls on finally statement e is true. The dimension of a plane is two on two is only one less than the dimension of our three. So in our three planes are also hyper planes

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