Linear Algebra

36 Chapter One. Linear Systemsand lines in even higher-dimensional spaces work in the same way.In R 3 , a line uses one parameter so that a particle on that line is free tomove back and forth in one dimension, and a plane involves two parameters.For example, the plane through the points (1, 0, 5), (2, 1, −3), and (−2, 4, 0.5)consists of (endpoints of) the vectors in⎛⎜1⎞ ⎛ ⎞ ⎛ ⎞1 −3⎟ ⎜ ⎟ ⎜ ⎟{ ⎝0⎠ + t ⎝ 1⎠ + s ⎝ 4⎠ ∣ t, s ∈ R}5 −8 −4.5(the column vectors associated with the parameters⎛ ⎞ ⎛ ⎞ ⎛1 2⎜ ⎟ ⎜ ⎟ ⎜1⎞ ⎛ ⎞ ⎛ ⎞ ⎛−3 −2⎟ ⎜ ⎟ ⎜ ⎟ ⎜1⎞⎟⎝ 1⎠ = ⎝ 1⎠ − ⎝0⎠⎝ 4⎠ = ⎝ 4⎠ − ⎝0⎠−8 −3 5 −4.5 0.5 5are two vectors whose whole bodies lie in the plane). As with the line, note thatwe describe some points in this plane with negative t’s or negative s’s or both.In algebra and calculus we often use a description of planes involving a singleequation as the condition that describes the relationship among the first, second,and third coordinates of points in a plane.⎛P = { ⎝ x ⎞y⎠ ∣ 2x + y + z = 4}zThe translation from such a description to the vector description that we favorin this book is to think of the condition as a one-equation linear system andparametrize x = 2 − y/2 − z/2.⎛P = { ⎝ x ⎞ ⎛y⎠ = ⎝ 2 ⎞ ⎛0⎠ + y · ⎝ −1/2 ⎞ ⎛1⎠ + z · ⎝ −1/2 ⎞0⎠ ∣ y, z ∈ R}z 001∣Generalizing, a set of the form {⃗p + t 1 ⃗v 1 + t 2 ⃗v 2 + · · · + t k ⃗v k t1 , . . . , t k ∈ R}where ⃗v 1 , . . . ,⃗v k ∈ R n and k n is a k-dimensional linear surface (or k-flat).For example, in R 4 ⎞ ⎞{⎛⎛2 1π⎜ ⎟⎝ 3⎠ + t 0⎜ ⎟⎝0⎠−0.5 0∣ t ∈ R}