Worksheet on linear combinations fall 2010 PDF

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Math 175 Fall 2010 Dr. Seelinger

Worksheet on Linear Combinations We make the following definition. Definition. A linear combination of vectors ~v1 , ~v2 , . . . , ~vk ∈ Rn is a vector w ~ ∈ Rn that can be written in the form w ~ = a1~v1 + a2~v2 + · · · + ak~vk for some scalars a1 , a2 , . . . , ak ∈ R.      −1 1 0  0   0   −1       1. Let ~v1 =   1  , ~v2 =  1  , ~v3 =  0 . Write a general expression for a linear combination 4 2 3 of ~v1 , ~v2 , and ~v3 . In this case, what value does the k of the definition take on? 

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5  4 2. Is w ~=  3 2 ~v2 , and ~v3 .

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  a linear combination of ~v1 , ~v2 , and ~v3 ? If so, write w ~ as a linear combination of ~v1 ,  If not, explain why not.

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 1  −5   ~ as a linear combination of 3. Is ~u =   9  a linear combination of ~v1 , ~v2 , and ~v3 ? If so, write w 35 ~v1 , ~v2 , and ~v3 . If not, explain why not. 

   0 1 −1 a1   −1 0 0  4. Let B =  . Find an expression for B  a2 . How does this compare with your  1 1 0 a3 3 2 4 answer to Question 1? 

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