Linear Algebra

36 Chapter One. Linear Systemsand lines in even higher-dimensional spaces work in the same way.If a line uses one parameter, so that there is freedom to move back andforth in one dimension, then a plane must involve two. For example, the planethrough the points (1, 0, 5), (2, 1, −3), and (−2, 4, 0.5) consists of (endpoints of)the vectors in⎛ ⎞ ⎛1{ ⎝0⎠ + t · ⎝511−8⎞⎛⎠ + s · ⎝−34−4.5⎞⎠ ∣ ∣ t, s ∈ R}(the column vectors associated with the parameters⎛⎝ 1 ⎞ ⎛1 ⎠ = ⎝ 2 ⎞ ⎛1 ⎠ − ⎝ 1 ⎞ ⎛0⎠⎝ −3⎞ ⎛ ⎞ ⎛−24 ⎠ = ⎝ 4 ⎠ − ⎝ 1 ⎞0⎠−8 −3 5 −4.5 0.5 5are two vectors whose whole bodies lie in the plane). As with the line, note thatsome points in this plane are described with negative t’s or negative s’s or both.A description of planes that is often encountered in algebra and calculususes a single equation as the condition that describes the relationship amongthe first, second, and third coordinates of points in a plane.P = {( xyz) ∣∣2x + y + z = 4}The 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 = (1/2)(4 − y − z).P = {( 200)+( ) ( )−0.5 −0.51 y + 0 z ∣ y, z ∈ R}01Generalizing from lines and planes, we define a k-dimensional ∣ linear surface(or k-flat) in R n to be {⃗p + t 1 ⃗v 1 + t 2 ⃗v 2 + · · · + t k ⃗v k t 1 , . . . , t k ∈ R} where⃗v 1 , . . . , ⃗v k ∈ R n . For example, in R 4 ,⎛{ ⎜⎝2π3−0.5⎞ ⎛ ⎞1⎟⎠ + t ⎜0⎟⎝0⎠0∣ t ∈ R}