# 1 Dimensions11 Prove that a set of 3 points a, b, and c in

1 Dimensions1:1 Prove that a set of 3 points a, b, and c in the plane are collinear if and only if the vectorsa1,b1, andc1are linearly dependent, i.e. one of these points can be written as a linear combination of the other two.Note, you should use the equation of the line through the points to dene the linear dependence.1:2 We say that ane dimension of a set of points is the dimension of the space of all ane combinations.A point is 0-dimensional. A line is 1-dimensional. A plane is 2-dimensional. This is a little dierent fromthe notion of dimension used for vector spaces. It is true that a 1-dimensional vector space is a line, butthat means it can be represented as all linear combinations of a single vector. For a general line, we need atleast two points. (This is the dierence between vectors and points). Prove that the ane dimension of aset of points p1; : : : ; pn 2 R3is one less than the dimension of the vector space spanned byp11; : : : ;pn1.1:3 Most of the time, question 1.2 means that the ane dimension of fp1; : : : ; png is one less than the lineardimension of the same set of points when viewed as vectors. When is the ane dimension equal to the lineardimension?Attachments: 687864_1_ggghw1.jpg

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