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1. Lines and Planes 1 Lines and Planes Linear algebra is the study of linearity in its most general algebraic forms. A collection of mathematical objects exhibit linearity • if, whenever X and Y are two such objects, they can be added in a well-defined manner (that is, X + Y makes sense as an object of the same type); • if for all numbers a, the scalar multiple aX makes sense as an object of the same type as X; and • for all such objects X and Y and numbers a, scalar multiplication distributes over addition: a(X + Y) = aX + aY. You have already experienced systems of € mathematical objects that behave linearly, i.e., according to the properties listed above. The prototypical examples of such systems are vectors in the Euclidean plane R 2 or vectors in Euclidean space R 3 (and, indeed, from any R n ). For this reason, we call any system that satisfies the defining properties above (even if it is not of the € n form R for some positive € € integer n) a vector space. € 1. Lines and Planes 2 The centrality of the examples of R 2 and R 3 derives from the fact that we have a geometric intuition about their objects. € € Vectors in R 2 can be visualized as arrows of a certain length and pointing in a certain direction (the zero vector being an exception in that it has zero length and no specified direction). This gives € us a mental picture of what it means to add vectors – via the parallelogram rule (see Figure 1.1.2, p. 2); or, to multiply a vector by a scalar, which either magnifies it – if the scalar has absolute value larger than 1 – or shrinks it – if the scalar has absolute value smaller than 1, while maintaining its direction – if the scalar is positive – or reversing direction – if the scalar is negative (see Figures 1.1.3, 1.1.4, p. 3). The picture that we attach to vectors in R 2 – and the situation in R 3 is much the same, with the added complexity of a third dimension – makes it clear why the subject is called linear algebra: in R 2 € (and in € R 3 ), we can characterize a line L in terms of vectors that satisfy the three properties we laid out at the beginning of this discussion. € € 1. Lines and Planes 3 Where P and Q are any fixed vectors whose tips lies on L, then Q – P is a vector that points in the direction of L (and this is true regardless how P and Q are chosen along L). Indeed, any two vectors V and W that point in the direction of L have a sum that also points in the direction of L; further, any scalar multiple of such a vector points in the direction of L, and all such vectors obey the distributive property of scalar multiplication over vector addition. Therefore, the set of scalar multiples of Q – P forms a vector space. It follows that all every vector X whose tip lies on L has the form X = P + t(Q – P) or X = (1 – t)P + tQ, for some scalar t. So if we let t run through all real values, these last equations are equivalent forms for parametrizing the line L. The variable scalar t here is called the parameter of the equation. 1. Lines and Planes € 4 In R 2 , we can write out these vectors in coordinate form: P = (p1, p2 ), Q = (q1,q2 ) are fixed vectors, and X = (x, y) is a variable vector that depends on the parameter t. Expanding via coordinates gives the system of Cartesian parametric equations € € € x = p1 + t(q1 − p1 ) y = p2 + t(q2 − p2 ) for the line L. Since Q ≠ P (why not?), at least one of the two coefficients of t above is not zero, so we can solve€for t in that equation and substitute that expression into the other equation; simplifying produces a single equation of the form ax + by + c = 0, the Cartesian equation of the line L in the plane. € In R 3 the algebra is entirely similar, except that we get a system of three Cartesian parametric equations of the form € x = p1 + t(q1 − p1 ) y = p2 + t(q2 − p2 ) z = p3 + t(q3 − p3 ) € 1. Lines and Planes 5 Eliminating the parameter as before produces a pair of equations of the form ax + by + cz + d = 0 a′x + b′y + c′z + d′ = 0 the Cartesian equations of the line L in space. € way we can describe equations of a In a similar plane Π in R 3 : where P, Q, R are fixed vectors whose tips lie on Π and are not collinear (that is, they don’t lie on the same line within Π), then Q – P and R – P are vectors that point in different € € directions within Π. € € It follows that any vector lying in Π can be expressed as a sum of some scalar multiple of Q – P € and some scalar multiple of R – P. € If X is any vector whose tip lies on Π, then X – P is a vector lying in Π, so X – P = s(Q – P) + t(R – P) for some pair of scalars s and t. Therefore, € € X = P + s(Q – P) + t(R – P), which parametrizes the vector equation of the plane; here, both s and t are variable parameters. (We could also write X = (1 – s – t)P + sQ + tR.) 1. Lines and Planes 6 Writing this out in coordinate form produces a system of Cartesian parametric equations for the plane Π of the form x = p1 + s(q1 − p1 ) + t(r1 − p1 ) € y = p2 + s(q2 − p2 ) + t(r2 − p2 ) z = p3 + s(q3 − p3 ) + t(r3 − p3 ) Finally, treating this as a system of three equations in the two unknowns s and t, we can use one of the € equations to solve for one of the two variables, substitute that expression into the other two equations, and thereby obtain a system of two equations in one unknown. Eliminating the remaining parameter will produce a single equation of the form ax + by + cz + d = 0 , the Cartesian equation of the plane Π. (Observe€again that the set of sums of scalar multiples of Q – P with scalar multiples of R – P € forms a vector space!)