c2s5 - 1 2.5 Lines, Planes, and Other Flats Chapter 2....

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2.5 Lines, Planes, and Other Flats 1 Chapter 2. Dimension, Rank, and Linear Transformations 2.5 Lines, Planes, and Other Flats Defnitions 2.4, 2.5. Let S be a subset of R n and let ~a R n .T h e set { ~x + ~a | ~x S } is the translate of S by ~a , and is denoted by S + ~a . The vector ~a is the translation vector .A line in R n is a translate of a one-dimensional subspace of R n
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2.5 Lines, Planes, and Other Flats 2 Defnition. If a line L in R n contains point ( a 1 ,a 2 ,...,a n ) and if vector ~ d is parallel to L ,then ~ d is a direction vector for L and ~a =[ a 1 ,a 2 ,...,a n ] is a translation vector of L . Note. With ~ d as a direction vector and ~a as a translation vector of a line, we have L = { t ~ d + ~a | t R } . In this case, t is called a parameter and we can express the line parametrically as a vector equation: ~x = t ~ d + ~a or as a collection of component equations: x 1 = td 1 + a 1 x 2 = td 2 + a 2 . . . x
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c2s5 - 1 2.5 Lines, Planes, and Other Flats Chapter 2....

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