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Math 115 Fall 2012 3/4 Course Notes

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Department
Mathematics
Course
MATH 115
Professor
Martin Pei
Semester
Fall

Description
MATH 115 - Linear Algebra for Engineers Kevin Carruthers Fall 2012 Vectors Vectors have a magnitude and a direction, are denoted by vector arrows, and are said to be in Rn Two Main Operations You can perform two main operations with vectors: vector addition, which is the algnbraic tail to tip addition of vectors, and scalar multiplication, whv where t R andv R Linear Combinations The linear combinatio of a set of vect~ ;:::~ is any vector which can be obtained from 0 n these vectors through vector addition and scalar multiplication. It has the0 0++v an nwhere a 0:::a n R 1 ~ 0 Example: for a = 0 and b = 1 ~ 3 3a 2b = 2 Any vector in R is a linear combination of this set. 1 Dot Product 2 3 2 3 a0 b0 In R , the dot product ou = 4::: and v =4 ::: is a scalar dened to be an bn a b = a b + + a b 0 0 n n thus ia = 3 and v = 2 5 6 u v = 3 2 + 5 6 = 36 Properties u v = v u~ tu v = tu ~v), for any scalar t u v + ~) = u v + u ~w Magnitude The magnitude of a vector is its length. 2 3 1 6 0 6 7 Example: for v =6 7 4 1 2 p v = 3 + 4 = 5 More generally p v = v v and 2 v = v v Unit Vector The unit vector is a vector of length one. Giv, the unit vector with the same direction is ~v 2 This is called normalization. Distance Between Points For P and Q, the distance between them is PQ Angle Between Two Vectors For u and v, the angle between them is . 2 2 2 Deriving from the cosine law c = a + b 2abcos, we get 2 2 2 u ~v = u + ~v 2 ~uv cos = (~u ~v) u ~v) = ~u u u v ~v u +v ~v = u 2u v + ~v2 2 ~uv cos = 2~u v cos = ~uv 1 = cos uv Denition: two vectors u and v are orthogonal iu ~v = 0 Lines In R ;y = mx + b, but in R we must two vectors to represent a line. The equation for a line in R follows the same forms as in R , but uses any vector on the line as its intercept, and any vector parallel to the line as its slope. Example: x = (3;1;4;1) + t(2;0;3;2);t R 3 Parametric Form In parametric form, we solve for the values of each variable in the resulting vector. For the above equation we have 8 x = 3 + 2t > 0 < x 1 1 x = > x 2 4 + 3t : x 3 1 + 2t where t R 3 Planes (In R only) 3 Every plane in R has a normal vector orthogonal to the plane.n must be orthogonal to PX, where P and X are both points on the plane, so n PX = 0 2 3 a For any line in R of the form ax + by + cz = n =~4b5 c Projections proj (v) is the projection ov ontou~ u 1. prou v) is a scalar multipleuof ~ 2. prou v) andv proju(v) are orthogonal u v projuv) = 2u u perp uv) = v projuv) perp (v) projv) = 0 u u Shortest Distance The tip of prouv) is the closest point tv ~ 4
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