section42 - Chapter 4 The Vector Space Rn MAT188H1F Lec03...

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Chapter 4: The Vector Space R n MAT188H1F Lec03 Burbulla Chapter 4 Lecture Notes Fall 2007 Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n Chapter 4: The Vector Space R n 4.2: Linear Independence Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 4: The Vector Space R n 4.2: Linear Independence Linear Independence Definition: Let S = { X 1 , X 2 , . . . , X k } be a set of k vectors in R n , let a 1 , a 2 , . . . , a k be scalars, and suppose a 1 X 1 + a 2 X 2 + · · · + a k X k = O . If the only solution for the scalars are a 1 = 0 , a 2 = 0 , . . . , a k = 0 , then the set S is called a linearly independent set, and we say that the vectors X 1 , X 2 , . . . , X k are linearly independent. Purely symbolically: a 1 X 1 + a 2 X 2 + · · · + a k X k = O a 1 = 0 , a 2 = 0 , . . . , a k = 0 means the vectors X 1 , X 2 , . . . , X k are linearly independent. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.2: Linear Independence Example 1 Suppose X 1 = 1 0 1 - 1 , X 2 = 3 2 1 - 1 , X 3 = - 1 0 4 5 . Then a 1 X 1 + a 2 X 2 + a 3 X 3 = O a 1 + 3 a 2 - a 3 = 0 2 a 2 = 0 a 1 + a 2 + 4 a 3 = 0 - a 1 - a 2 + 5 a 3 = 0 Reducing the augmented matrix: 1 3 - 1 0 0 2 0 0 1 1 4 0 - 1 - 1 5 0 1 3 - 1 0 0 1 0 0 0 1 1 0 0 - 1 8 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 . This means a 1 = 0 , a 2 = 0 , a 3 = 0 , so the vectors X 1 , X 2 , X 3 are linearly independent. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 4: The Vector Space R n 4.2: Linear Independence Example 2 Suppose X 1 = 1 0 1 - 1 , X 2 = 3 2 1 - 1 , X 3 = 1 2 - 1 1 . Then a 1 X 1 + a 2 X 2 + a 3 X 3 = O a 1 + 3 a 2 + a 3 = 0 2 a 2 + 2 a 3 = 0 a 1 + a 2 - a 3 = 0 - a 1 - a 2 + a 3 = 0 Reducing the augmented matrix: 1 3 1 0 0 2 2 0 1 1 - 1 0 - 1 - 1 1 0 1 3 1 0 0 1 1 0 0 - 2 - 2 0 0 2 2 0 1 0 - 2 0 0 1 1 0 0 0 0 0 0 0 0 0 . One non-zero solution is a 1 = 2 , a 2 = - 1 , a 3 = 1 , so the vectors X 1 , X 2 , X 3 are not linearly independent. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.2: Linear Independence Linear Dependence Definition: If S = { X 1 , X 2 , . . . , X k } is not linearly independent, it is called linearly dependent, and we say that the vectors X 1 , X 2 , . . . , X k are linearly dependent. That is, there are scalars
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