M341-1.1 - Multiplication of Vectors by scalars(Scalar...

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Multiplication of Vectors by scalars (Scalar Multiplication): R × R n R n ( c, x ) 7→ c x c, x 1 x 2 . . x n · 7→ cx 1 cx 2 . . cx n Addition of Vectors: R n × R n R n ( x , y ) 7→ x + y x 1 x 2 . . x n , y 1 y 2 . . y n · 7→ x 1 + y 1 x 2 + y 2 . . x n + y n Properties of these Operations: see Theorem 1.3 page 10. These properties are consequences of the fact that in each coordi- nate we multiply or add real numbers. These operations inherit of the properties of addition and multiplication of real numbers. 0 is neutral (rather than identity) for addition, - x is the opposite of x (rather than inverse for addition). These properties give sense to Linear Combinations. Exercises to make sure you understand the definitions: 7 , 8 , 6 . 1
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Dot (Inner) Product: R n × R n R ( x , y ) 7→ x . y x 1 x 2 . . x n , y 1 y 2 . . y n · 7→ x 1 y 1 + ... + x n y n Properties of the Dot Product: see Theorem 1.5 page 17.
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