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Math 70 sec1_3ov

# Math 70 sec1_3ov - 1.3 VECTOR EQUATIONS Key concepts to...

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1.3 VECTOR EQUATIONS Key concepts to master: linear combinations of vectors and a spanning set. Vector: A matrix with only one column. Vectors in R n (vectors with n entries): u u 1 u 2 u n Geometric Description of R 2 Vector x 1 x 2 is the point x 1 , x 2 in the plane. R 2 is the set of all points in the plane. 1

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Parallelogram rule for addition of two vectors: If u and v in R 2 are represented as points in the plane, then u v corresponds to the fourth vertex of the parallelogram whose other vertices are 0 , u and v . (Note that 0 0 0 .) EXAMPLE: Let u 1 3 and v 2 1 . Graphs of u , v and u v are given below: 1 2 3 4 x 1 1 2 3 4 x 2 Illustration of the Parallelogram Rule 2
EXAMPLE: Let u 1 2 . Express u , 2 u , and 3 2 u on a graph. - 2 - 1 1 2 x 1 - 3 - 2 - 1 1 2 3 4 x 2 3

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Linear Combinations DEFINITION Given vectors v 1 , v 2 , , v p in R n and given scalars c 1 , c 2 , , c p , the vector y defined by y c 1 v 1 c 2 v 2 c p v p is called a linear combination of v 1 , v 2 , , v p using weights c 1 , c 2 , , c p .
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