02-21-03 - Lecture 9 Andrei Antonenko February 21, 2003 1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 4

02-21-03 - Lecture 9 Andrei Antonenko February 21, 2003 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online