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Unformatted text preview: Lecture 9 Andrei Antonenko February 21, 2003 1 Linear combinations Definition 1.1. Let V be a vector space. A vector v ∈ V is a linear combination of vectors u 1 ,u 2 ,...,u n if there exist a 1 ,a 2 ,...,a n ∈ k such that v = a 1 u 1 + a 2 u 2 + ··· + a n u n . (1) Sometimes it is possible to express a vector as a linear combination of other vectors. It can be done by solving a corresponding linear system. We’ll demonstrate it in the following example. Example 1.2. Consider the space R 2 — the space of all pairs of numbers. Let v = (8 , 13) , u 1 = (1 , 2) , and u 2 = (2 , 3) . Let’s express v as a linear combination of u 1 and u 2 . To do this we have to find a and b such that v = au 1 + bu 2 , i.e. (8 , 13) = a (1 , 2)+ b (2 , 3) = ( a · 1+ b · 2 ,a · 2+ b · 3) . So, we get the following system: ( 1 a + 2 b = 8 2 a + 3 b = 13 We can simply solve this system: subtracting the first equation multiplied by 2 from the second one we get b = 3 , so b = 3 , and so a = 8 2 b = 2 . So we see that (8 , 13) = 2 · (1 , 2)+3 · (2 , 3) . Example 1.3. Consider the space P ( t ) — space of all polynomials. Let v = 5 t 2 + 2 t + 1 , u 1 = t 2 + t , u 2 = t + 1 , u 3 = t 2 + 1 . Let’s express v as a linear combination of u 1 , u 2 and u 3 . We should find a , b and c such that v = au 1 + bu 2 + cu 3 , i.e. 5 t 2 + 2 t + 1 = a ( t 2 + t ) + b ( t + 1) + c ( t 2 + 1) = t 2 ( a + c ) + t ( a + b ) + ( b...
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.
 Spring '03
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