CHAPTER 4 Vector Spaces Sec 4 1 The

CHAPTER 4 Vector Spaces Sec 4. 1 The Vector Space R 3 Three-dim coordinate space

Sec 4. 1 The Vector Space R 3 Three-dim coordinate space z P(a, b, c) v y We call the number a , b , c the components of the vector v The point P(a, b, c) determines the vector v=(a, b, c) The position vector c a x b

Sec 4. 1 The Vector Space R 3 Addition of vectors Multiplication of a vector by a scalar

Sec 4. 1 The Vector Space R 3 The length of a vector z P(a, b, c) is defined to be the distance of the point p(a, b, c) from the origin v Find c y a The geometric interpretation x b C>0, The vector c v is of length c(length v) of the same direction C<0, The vector c v is of length c(length v) of the opposite direction

Sec 4. 1 The Vector Space R The geometric interpretation Triangle law of addition Parallelogram law of addition 3

Sec 4. 1 The Vector Space R Vector Space Set S with vector addition and multiplication by scalar is a vector space if these operations satisfy the following: S is a vector space 3

Sec 4. 1 The Vector Space R 3 Addition of vectors Multiplication of a vector by a scalar

Sec 4. 1 The Vector Space R 3 Linear combination By solving the linear system W 1 is a linear combination of u 1, u 2, u 3 W 4 is a linear combination of u 1, u 2, u 3

Sec 4. 1 The Vector Space R Linear combination Cab w 1 be written as a linear combination of u 1, u 2, u 3 3

Sec 4. 1 The Vector Space R 3 Linearly dependent vectors are said to be linearly dependent provided that one of them is a linear combination of the other two

Sec 4. 1 The Vector Space R 3 Linearly dependent vectors are said to be linearly dependent provided that one of them is a linear combination of the other two otherwise, they are linearly independent TH 3: the three vectors u 1, u 2, u 3 are linearly dependent iff there exist scalars c 1, c 2, c 3 not all zeros such that NOTE: if the only solution of (*) is c 1=0, c 2=0, c 3=0 then u 1, u 2, u 3 are linearly independent

Sec 4. 1 The Vector Space R 3 Linearly dependent vectors NOTE: if the only solution of (*) is c 1=0, c 2=0, c 3=0 then u 1, u 2, u 3 are linearly independent TH 4: the three vectors u 1, u 2, u 3 are linearly independent iff

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