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 u + v is given by placing the start of v where u ends. Check this by drawing u = 1 3 , v = 2 1 , and u + v = 1 + 2 3 + 1 = 3 4 .
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