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# m235notes3 - Linear Algebra and Geometry We construct a...

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Linear Algebra and Geometry We construct a dictionary between some geometrical notions and some notions from linear algebra. Adding, Scalar Multiplication An element of ( x,y ) R 2 corresponds to an arrow with tail at the origin in R 2 and head at the point ( x,y ). Two arrows are the same vector if they have the same length and direction. This works the same way in R 3 or R n for any n . We can add vectors algebraically. We have ( x 1 ,y 1 ) + ( x 2 ,y 2 ) = ( x 1 + x 2 ,y 1 + y 2 ). This corresponds to the geometrical operation of placing the vector ( x 2 ,y 2 ) so that its tail is at the head of ( x 1 ,y 1 ). The result of adding is the vector with tail at the origin and head at the head of ( x 2 ,y 2 ). We can interchange the roles of the two vectors and we obtain the same result. Addition works the same in R 3 . We can multiply a vector ( x,y ) by a scalar a R . Algebraically we get a ( x,y ) = ( ax,ay ). Geometrically this is the operation of stretching or squeezing the vector ( x,y ) by a factor of a if a 0. If a < 0, then we turn the vector around and stretch by a factor of | a | . We can combine the two operations. Let ~u = ± 1 2 ² and ~v = ± 3 1 ² . What is the geometrical picture of 2 ~u - v ? 1

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How do we visualize this? Start by sketching examples such as 2 ± 1 2 ² + 3 ± - 3 1 ² , or - 2 ± 1 2 ² + 2 ± - 3 1 ² . The Geometry of Solving Equations in the Plane We can draw a picture of solving linear equations. We can look at each of the equations x - 3 y = 1 2 x + y = - 5 as deﬁning a line, so the solution to the set of equations is the intersection of the lines. A single linear equation in variables x,y can be thought of as eliminating all the points in the plane except those satisfying the equation. This description allows us to see what happens when we are given two linear equations in two unknowns. Three diﬀerent things can happen. The two corresponding lines can intersect in one point, the unique solution to the
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## This note was uploaded on 11/22/2011 for the course MATH 235 taught by Professor Markman during the Fall '08 term at UMass (Amherst).

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m235notes3 - Linear Algebra and Geometry We construct a...

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