l3 - Math 20F Linear Algebra Lecture 3: Vector Equations,...

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Math 20F Linear Algebra Lecture 3: Vector Equations, Linear Combinations and Span. In linear algebra we think of vectors in R n as column vectors or n × 1 matrices u = u 1 u 2 . . . u n , v = v 1 v 2 . . . v n Addition and scalar multiplication are defined by u + v = u 1 + v 1 u 2 + v 2 . . . u n + v n , λ u = λu 1 λu 2 . . . λu n , λ R . Linear Combination : Given vectors v 1 ,..., v k , and scalars λ 1 ,...,λ k , the vector w = λ 1 v 1 + ··· + λ k v k is called a linear combination of the vectors v 1 ,..., v k , (with weights λ 1 ,...λ k ). Question : If you give me a vector w , and vectors v 1 ,..., v k , how can I figure out whether w is a linear combination of v 1 ,... v k ? Geometric Interpretation : In R 2 and R 3 We think of vectors as arrows with a length and a direction. The parallelogram law says that the sum
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This note was uploaded on 04/29/2008 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.

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l3 - Math 20F Linear Algebra Lecture 3: Vector Equations,...

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