one_ii.pdf - One.II Linear Geometry Linear Algebra Jim...

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One.II Linear Geometry Linear Algebra Jim Hefferon
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Geometry We can draw two-unknown equations as lines. Then the three possibilities for solution sets become clear.
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Geometry We can draw two-unknown equations as lines. Then the three possibilities for solution sets become clear. Unique solution 3x + 2y = 7 x - y = - 1 No solutions 3x + 2y = 7 3x + 2y = 4 Infinitely many solutions 3x + 2y = 7 6x + 4y = 14 Besides being pretty, the geometry helps us understand what is happening.
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Vectors in space
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Vectors A vector is an object consisting of a magnitude and a direction. For instance, a vector can model a displacement.
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Vectors A vector is an object consisting of a magnitude and a direction. For instance, a vector can model a displacement. Two vectors with the same magnitude and same direction, such as all of these, are equal. For instance, each of the above could model a displacement of one over and two up.
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Denote the vector that extends from ( a 1 , a 2 ) to ( b 1 , b 2 ) by b 1 - a 1 b 2 - a 2 so the “one over, two up” vector would be written in this way. 1 2
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Denote the vector that extends from ( a 1 , a 2 ) to ( b 1 , b 2 ) by b 1 - a 1 b 2 - a 2 so the “one over, two up” vector would be written in this way. 1 2 We often picture a vector ~ v = v 1 v 2 as starting at the origin. From there ~ v extends to ( v 1 , v 2 ) and we may refer to it as “the point ~ v ” so that we may call each of these R 2 . { ( x 1 , x 2 ) | x 1 , x 2 R } { x 1 x 2 | x 1 , x 2 R }
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These definitions extend to higher dimensions. The vector that starts at ( a 1 , . . . , a n ) and ends at ( b 1 , . . . , b n ) is represented by this column b 1 - a 1 . . . b n - a n and two vectors are equal if they have the same representation. Also, we aren’t too careful about distinguishing between a point and the vector which, when it starts at the origin, ends at that point.
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